id stringlengths 11 15 | title stringlengths 7 171 | problem stringlengths 9 4.33k | solution stringlengths 6 19k | problem_wikitext stringlengths 9 4.42k | solution_wikitext stringlengths 7 19.1k | proof_title stringlengths 9 171 | theorem_url stringlengths 34 198 | proof_url stringlengths 36 198 | categories listlengths 0 9 | theorem_references listlengths 0 36 | proof_references listlengths 0 253 |
|---|---|---|---|---|---|---|---|---|---|---|---|
proofwiki-23500 | Fully T4 Space is Paracompact | Let $T = \struct {S, \tau}$ be a fully $T_4$ space.
Then $T$ is paracompact. | {{Recall|Paracompact Space|paracompact}}
{{:Definition:Paracompact Space}}
Let $T = \struct {S, \tau}$ be a fully $T_4$ space.
{{Recall|Fully T4 Space|fully $T_4$}}
{{:Definition:Fully T4 Space}}
Let $\UU$ be an open cover for $T$.
Then, from the definition of barycentric refinement, there exists a cover $\VV$ for $T$ ... | Let $T = \struct {S, \tau}$ be a [[Definition:Fully T4 Space|fully $T_4$ space]].
Then $T$ is [[Definition:Paracompact Space|paracompact]]. | {{Recall|Paracompact Space|paracompact}}
{{:Definition:Paracompact Space}}
Let $T = \struct {S, \tau}$ be a [[Definition:Fully T4 Space|fully $T_4$ space]].
{{Recall|Fully T4 Space|fully $T_4$}}
{{:Definition:Fully T4 Space}}
Let $\UU$ be an [[Definition:Open Cover|open cover]] for $T$.
Then, from the definition of... | Fully T4 Space is Paracompact | https://proofwiki.org/wiki/Fully_T4_Space_is_Paracompact | https://proofwiki.org/wiki/Fully_T4_Space_is_Paracompact | [
"Fully T4 Spaces",
"Paracompact Spaces",
"Sequence of Implications of Paracompactness Properties"
] | [
"Definition:Fully T4 Space",
"Definition:Paracompact Space"
] | [
"Definition:Fully T4 Space",
"Definition:Open Cover",
"Definition:Barycentric Refinement",
"Definition:Cover of Set",
"Fully Normal Space is Paracompact"
] |
proofwiki-23501 | Matrix is Unitary iff Columns are Orthonormal Basis | Let $\mathbf U$ be an $n \times n$ square matrix over $\mathbb C$.
Then:
:$\mathbf U$ is a unitary matrix
{{Iff}}
:The columns of $\mathbf U$ form an orthonormal basis of $\mathbb C^n$. | === Sufficient Condition ===
Let $\mathbf U$ be unitary.
Then:
{{begin-eqn}}
{{eqn | l = \mathbf U^\dagger
| r = \mathbf U^{-1}
| c = {{Defof|Unitary Matrix}}
}}
{{eqn | ll= \leadsto
| l = \mathbf U^\dagger \mathbf U
| r = \mathbf U^{-1} \mathbf U
}}
{{eqn | ll= \leadsto
| l = \mathbf U^\d... | Let $\mathbf U$ be an $n \times n$ [[Definition:Square Matrix|square matrix]] over $\mathbb C$.
Then:
:$\mathbf U$ is a [[Definition:Unitary Matrix|unitary matrix]]
{{Iff}}
:The [[Definition:Column of Matrix|columns]] of $\mathbf U$ form an [[Definition:Orthonormal Basis of Vector Space|orthonormal basis]] of $\mathbb... | === Sufficient Condition ===
Let $\mathbf U$ be [[Definition:Unitary Matrix|unitary]].
Then:
{{begin-eqn}}
{{eqn | l = \mathbf U^\dagger
| r = \mathbf U^{-1}
| c = {{Defof|Unitary Matrix}}
}}
{{eqn | ll= \leadsto
| l = \mathbf U^\dagger \mathbf U
| r = \mathbf U^{-1} \mathbf U
}}
{{eqn | ll= \... | Matrix is Unitary iff Columns are Orthonormal Basis | https://proofwiki.org/wiki/Matrix_is_Unitary_iff_Columns_are_Orthonormal_Basis | https://proofwiki.org/wiki/Matrix_is_Unitary_iff_Columns_are_Orthonormal_Basis | [
"Unitary Matrices",
"Orthonormal Bases"
] | [
"Definition:Matrix/Square Matrix",
"Definition:Unitary Matrix",
"Definition:Matrix/Column",
"Definition:Orthonormal Basis of Vector Space"
] | [
"Definition:Unitary Matrix",
"Definition:Matrix/Element",
"Definition:Matrix",
"Definition:Equation",
"Definition:Matrix/Row",
"Definition:Matrix/Column",
"Definition:Matrix/Column",
"Definition:Orthonormal Basis of Vector Space",
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Column",
"D... |
proofwiki-23502 | Diagonal Functor on Product Category has Right Adjoint Iff Category has Binary Products/Lemma 1 | Let $a: X \to Y, b_1 : Y \to Z_1, b_2 : Y \to Z_2 \in \mathbf C$ be morphisms.
::<nowiki>$\begin{xy}\xymatrix@+1em@L+3px{
&
X
\ar[d]_*+{a}
\\
&
Y
\ar[ld]_*+{b_1}
\ar[rd]^*+{b_2}
\\
Z_1
&
&
Z_2
}\end{xy}$</nowiki>
Then:
:$\family{b_1 \circ a, b_2 \circ a} = \family{b_1, b_2} \circ a$ | Consider the diagram:
::<nowiki>$\begin{xy}\xymatrix@+1em@L+3px{
&&
X
\ar[ldld]_*+{b_1 \circ a}
\ar[d]^*+{a}
\ar[rdrd]^*+{b_2 \circ a}
\\
&&
Y
\ar[ldl]_*+{b_1}
\ar@{-->}[d]^*+{\family{b_1, b_2} }
\ar[rdr]^*+{b_2}
\\
Z_1
&&
Z_1 \times Z_2
\ar[ll]^*+{p_1}
\ar[rr]_*+{p_2}
&&
Z_2
}\end{xy}$</nowiki>
Th... | Let $a: X \to Y, b_1 : Y \to Z_1, b_2 : Y \to Z_2 \in \mathbf C$ be [[Definition:Morphism (Category Theory)|morphisms]].
::<nowiki>$\begin{xy}\xymatrix@+1em@L+3px{
&
X
\ar[d]_*+{a}
\\
&
Y
\ar[ld]_*+{b_1}
\ar[rd]^*+{b_2}
\\
Z_1
&
&
Z_2
}\end{xy}$</nowiki>
Then:
:$\family{b_1 \circ a, b_2 \circ a} = \family... | Consider the diagram:
::<nowiki>$\begin{xy}\xymatrix@+1em@L+3px{
&&
X
\ar[ldld]_*+{b_1 \circ a}
\ar[d]^*+{a}
\ar[rdrd]^*+{b_2 \circ a}
\\
&&
Y
\ar[ldl]_*+{b_1}
\ar@{-->}[d]^*+{\family{b_1, b_2} }
\ar[rdr]^*+{b_2}
\\
Z_1
&&
Z_1 \times Z_2
\ar[ll]^*+{p_1}
\ar[rr]_*+{p_2}
&&
Z_2
}\end{xy}$</nowiki>
... | Diagonal Functor on Product Category has Right Adjoint Iff Category has Binary Products/Lemma 1 | https://proofwiki.org/wiki/Diagonal_Functor_on_Product_Category_has_Right_Adjoint_Iff_Category_has_Binary_Products/Lemma_1 | https://proofwiki.org/wiki/Diagonal_Functor_on_Product_Category_has_Right_Adjoint_Iff_Category_has_Binary_Products/Lemma_1 | [
"Diagonal Functor on Product Category has Right Adjoint Iff Category has Binary Products"
] | [
"Definition:Morphism"
] | [
"Definition:Commutative Diagram",
"Definition:Binary Product UMP (Category Theory)",
"Definition:Binary Product UMP (Category Theory)",
"Definition:Commutative Diagram",
"definition:Unique",
"Definition:Morphism",
"Category:Diagonal Functor on Product Category has Right Adjoint Iff Category has Binary P... |
proofwiki-23503 | Diagonal Functor on Product Category has Right Adjoint Iff Category has Binary Products/Lemma 2 | Let $a_1: X \to Y_1, a_2: X \to Y_2, b_1 : Y_1 \to Z_1, b_2 : Y_2 \to Z_2 \in \mathbf C$ be morphisms:
::<nowiki>$\begin{xy}\xymatrix@+1em@L+3px{
&
X
\ar[ld]_*+{a_1}
\ar[rd]^*+{a_2}
\\
Y_1
\ar[d]_*+{b_1}
&&
Y_2
\ar[d]^*+{b_2}
\\
Z_1
&&
Z_2
}\end{xy}$</nowiki>
Then:
:$\family{b_1 \circ a_1, b_2 \circ a_2} =... | Consider the diagram:
::<nowiki>$\begin{xy}
<0em,5em>*+{X} = "TM",
<-6em,0em>*+{Y_1} = "ML",
<0em,0em>*+{Y_1 \times Y_2} = "MM",
<6em,0em>*+{Y_2} = "MR",
<-6em,-5em>*+{Z_1} = "BL",
<0em,-5em>*+{Z_1 \times Z_2} = "BM",
<6em,-5em>*+{Z_2} = "BR",
"TM";"ML" **\crv{<-4em,5em>} ?>*@{>} ?<>(.8)*!/^.8em/{a_1},
"TM";"MM" **@{--... | Let $a_1: X \to Y_1, a_2: X \to Y_2, b_1 : Y_1 \to Z_1, b_2 : Y_2 \to Z_2 \in \mathbf C$ be [[Definition:Morphism (Category Theory)|morphisms]]:
::<nowiki>$\begin{xy}\xymatrix@+1em@L+3px{
&
X
\ar[ld]_*+{a_1}
\ar[rd]^*+{a_2}
\\
Y_1
\ar[d]_*+{b_1}
&&
Y_2
\ar[d]^*+{b_2}
\\
Z_1
&&
Z_2
}\end{xy}$</nowiki>
T... | Consider the diagram:
::<nowiki>$\begin{xy}
<0em,5em>*+{X} = "TM",
<-6em,0em>*+{Y_1} = "ML",
<0em,0em>*+{Y_1 \times Y_2} = "MM",
<6em,0em>*+{Y_2} = "MR",
<-6em,-5em>*+{Z_1} = "BL",
<0em,-5em>*+{Z_1 \times Z_2} = "BM",
<6em,-5em>*+{Z_2} = "BR",
"TM";"ML" **\crv{<-4em,5em>} ?>*@{>} ?<>(.8)*!/^.8em/{a_1},
"TM";"MM" **@... | Diagonal Functor on Product Category has Right Adjoint Iff Category has Binary Products/Lemma 2 | https://proofwiki.org/wiki/Diagonal_Functor_on_Product_Category_has_Right_Adjoint_Iff_Category_has_Binary_Products/Lemma_2 | https://proofwiki.org/wiki/Diagonal_Functor_on_Product_Category_has_Right_Adjoint_Iff_Category_has_Binary_Products/Lemma_2 | [
"Diagonal Functor on Product Category has Right Adjoint Iff Category has Binary Products"
] | [
"Definition:Morphism"
] | [
"Definition:Commutative Diagram",
"Definition:Binary Product UMP (Category Theory)",
"Definition:Product of Morphisms",
"Definition:Binary Product UMP (Category Theory)",
"Definition:Commutative Diagram",
"definition:Unique",
"Definition:Morphism",
"Category:Diagonal Functor on Product Category has Ri... |
proofwiki-23504 | Compact Space is Strongly Paracompact | Let $T = \struct {S, \tau}$ be a compact topological space.
Then $T$ is strongly paracompact. | {{Recall|Strongly Paracompact Space|strongly paracompact space}}
{{:Definition:Strongly Paracompact Space}}
Let $T = \struct {S, \tau}$ be a compact topological space.
{{Recall|Compact Topological Space|compact topological space}}
{{:Definition:Compact Topological Space/Definition 1}}
Let $\CC$ be an arbitrary open cov... | Let $T = \struct {S, \tau}$ be a [[Definition:Compact Topological Space|compact topological space]].
Then $T$ is [[Definition:Strongly Paracompact Space|strongly paracompact]]. | {{Recall|Strongly Paracompact Space|strongly paracompact space}}
{{:Definition:Strongly Paracompact Space}}
Let $T = \struct {S, \tau}$ be a [[Definition:Compact Topological Space|compact topological space]].
{{Recall|Compact Topological Space|compact topological space}}
{{:Definition:Compact Topological Space/Defini... | Compact Space is Strongly Paracompact | https://proofwiki.org/wiki/Compact_Space_is_Strongly_Paracompact | https://proofwiki.org/wiki/Compact_Space_is_Strongly_Paracompact | [
"Compact Topological Spaces",
"Strongly Paracompact Spaces",
"Sequence of Implications of Paracompactness Properties"
] | [
"Definition:Compact Topological Space",
"Definition:Strongly Paracompact Space"
] | [
"Definition:Compact Topological Space",
"Definition:Arbitrary",
"Definition:Open Cover",
"Definition:Compact Topological Space",
"Definition:Subcover/Finite",
"Definition:Open Refinement",
"Finite Cover is Star-Finite",
"Definition:Star-Finite",
"Definition:Arbitrary",
"Definition:Open Cover",
"... |
proofwiki-23505 | Finite Cover is Star-Finite | Let $T = \struct {S, \tau}$ be a topological space.
Let $\CC$ be a finite cover for $T$.
Then $\CC$ is star-finite. | {{Recall|Star-Finite|star-finite}}
{{:Definition:Star-Finite}}
Let $\CC$ be a finite cover for $T$.
Let $H \in \CC$.
As $\CC$ is a finite set, it follows trivially that only a finite number of elements of $\CC$ intersect $H$.
Hence the result.
{{qed}}
Category:Star-Finite
Category:Covers
73eh4e2pdx5dj2hdke1fdb2mswybmdh | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $\CC$ be a [[Definition:Finite Cover|finite cover]] for $T$.
Then $\CC$ is [[Definition:Star-Finite|star-finite]]. | {{Recall|Star-Finite|star-finite}}
{{:Definition:Star-Finite}}
Let $\CC$ be a [[Definition:Finite Cover|finite cover]] for $T$.
Let $H \in \CC$.
As $\CC$ is a [[Definition:Finite Set|finite set]], it follows trivially that only a [[Definition:Finite Set|finite number]] of [[Definition:Element|elements]] of $\CC$ [[D... | Finite Cover is Star-Finite | https://proofwiki.org/wiki/Finite_Cover_is_Star-Finite | https://proofwiki.org/wiki/Finite_Cover_is_Star-Finite | [
"Star-Finite",
"Covers"
] | [
"Definition:Topological Space",
"Definition:Cover of Set/Finite",
"Definition:Star-Finite"
] | [
"Definition:Cover of Set/Finite",
"Definition:Finite Set",
"Definition:Finite Set",
"Definition:Element",
"Definition:Set Intersection",
"Category:Star-Finite",
"Category:Covers"
] |
proofwiki-23506 | Locally Finite Refinement is Point Finite | Let $\CC$ be a cover of a topological space $T = \struct {S, \tau}$.
Let $\UU$ be a locally finite refinement of $\CC$.
Then $\UU$ is also a point finite refinement of $\CC$. | {{Recall|Locally Finite Cover|locally finite cover}}
{{:Definition:Locally Finite Cover}}
{{Recall|Point Finite Cover|point finite cover}}
{{:Definition:Point Finite Cover}}
Let $\UU$ be a locally finite refinement of a cover $\CC$ of $S$.
Let $x \in S$.
Then there exists some neighborhood $N_x$ of $x$ which intersects... | Let $\CC$ be a [[Definition:Cover of Set|cover]] of a [[Definition:Topological Space|topological space]] $T = \struct {S, \tau}$.
Let $\UU$ be a [[Definition:Locally Finite Cover|locally finite]] [[Definition:Refinement of Cover|refinement]] of $\CC$.
Then $\UU$ is also a [[Definition:Point Finite Cover|point finite]... | {{Recall|Locally Finite Cover|locally finite cover}}
{{:Definition:Locally Finite Cover}}
{{Recall|Point Finite Cover|point finite cover}}
{{:Definition:Point Finite Cover}}
Let $\UU$ be a [[Definition:Locally Finite Cover|locally finite]] [[Definition:Refinement of Cover|refinement]] of a [[Definition:Cover of Set|c... | Locally Finite Refinement is Point Finite | https://proofwiki.org/wiki/Locally_Finite_Refinement_is_Point_Finite | https://proofwiki.org/wiki/Locally_Finite_Refinement_is_Point_Finite | [
"Refinements of Covers",
"Locally Finite Covers",
"Point Finite Covers"
] | [
"Definition:Cover of Set",
"Definition:Topological Space",
"Definition:Locally Finite Cover",
"Definition:Refinement of Cover",
"Definition:Point Finite Cover",
"Definition:Refinement of Cover"
] | [
"Definition:Locally Finite Cover",
"Definition:Refinement of Cover",
"Definition:Cover of Set",
"Definition:Neighborhood (Topology)/Point",
"Definition:Set Intersection",
"Definition:Finite Set",
"Definition:Element",
"Definition:Finite Set",
"Definition:Element",
"Definition:Refinement of Cover",... |
proofwiki-23507 | Submetacompact Space is Countably Metacompact | Let $T = \struct {S, \tau}$ be a submetacompact topological space.
Then $T$ is countably metacompact. | {{Recall|Countably Metacompact Space|countably metacompact space}}
{{:Definition:Countably Metacompact Space}}
Let $T = \struct {S, \tau}$ be a submetacompact space.
{{Recall|Submetacompact Space|submetacompact space}}
{{:Definition:Submetacompact Space}}
{{ProofWanted}} | Let $T = \struct {S, \tau}$ be a [[Definition:Submetacompact Space|submetacompact topological space]].
Then $T$ is [[Definition:Countably Metacompact Space|countably metacompact]]. | {{Recall|Countably Metacompact Space|countably metacompact space}}
{{:Definition:Countably Metacompact Space}}
Let $T = \struct {S, \tau}$ be a [[Definition:Submetacompact Space|submetacompact space]].
{{Recall|Submetacompact Space|submetacompact space}}
{{:Definition:Submetacompact Space}}
{{ProofWanted}} | Submetacompact Space is Countably Metacompact | https://proofwiki.org/wiki/Submetacompact_Space_is_Countably_Metacompact | https://proofwiki.org/wiki/Submetacompact_Space_is_Countably_Metacompact | [
"Submetacompact Spaces",
"Countably Metacompact Spaces",
"Sequence of Implications of Paracompactness Properties"
] | [
"Definition:Submetacompact Space",
"Definition:Countably Metacompact Space"
] | [
"Definition:Submetacompact Space"
] |
proofwiki-23508 | Pseudometrizable Space has Sigma-Locally Finite Basis | Let $T = \struct {S, \tau}$ be a pseudometrizable topological space.
Then $T$ has a basis that is $\sigma$-locally finite. | === Construction of Basis $\VV$ ===
We construct a $\sigma$-locally finite basis $\VV$.
For each $n \in \N$, let:
:$\UU_n = \set {\map {B_{1 / 2^n} } x : x \in S}$
That is, $\UU_n$ is the set of all open balls of radius $\dfrac 1 {2^n}$.
From Open Balls of Same Radius form Open Cover:
:$\forall n \in \N: \UU_n$ is an o... | Let $T = \struct {S, \tau}$ be a [[Definition:Pseudometrizable Space|pseudometrizable topological space]].
Then $T$ has a [[Definition:Basis (Topology)|basis]] that is [[Definition:Sigma-Locally Finite Basis|$\sigma$-locally finite]]. | === Construction of Basis $\VV$ ===
We construct a [[Definition:Sigma-Locally Finite Set of Subsets|$\sigma$-locally finite]] [[Definition:Basis (Topology)|basis]] $\VV$.
For each $n \in \N$, let:
:$\UU_n = \set {\map {B_{1 / 2^n} } x : x \in S}$
That is, $\UU_n$ is the [[Definition:Set|set]] of all [[Definition:Op... | Pseudometrizable Space has Sigma-Locally Finite Basis | https://proofwiki.org/wiki/Pseudometrizable_Space_has_Sigma-Locally_Finite_Basis | https://proofwiki.org/wiki/Pseudometrizable_Space_has_Sigma-Locally_Finite_Basis | [
"Pseudometrizable Topologies",
"Sigma-Locally Finite Bases",
"Nagata-Smirnov Metrization Theorem"
] | [
"Definition:Pseudometrizable Topology",
"Definition:Basis (Topology)",
"Definition:Sigma-Locally Finite Basis"
] | [
"Definition:Sigma-Locally Finite Set of Subsets",
"Definition:Basis (Topology)",
"Definition:Set",
"Definition:Open Ball",
"Definition:Open Ball/Radius",
"Open Balls of Same Radius form Open Cover",
"Definition:Open Cover",
"Metric Space is Paracompact",
"Definition:Paracompact Space",
"Definition... |
proofwiki-23509 | Hermitian Conjugate of Unitary Matrix is Unitary | The Hermitian conjugate of a unitary matrix is unitary. | Let $\mathbf U$ be unitary.
Then:
{{begin-eqn}}
{{eqn | l = \mathbf U
| r = \mathbf U
}}
{{eqn | ll=\leadsto
| l = \paren {\mathbf U^\dagger}^\dagger
| r = \paren {\mathbf U^{-1} }^{-1}
| c = Hermitian Conjugate is Involution
}}
{{eqn | ll=\leadsto
| l = \paren {\mathbf U^\dagger}^\dagger
... | The [[Definition:Hermitian Conjugate|Hermitian conjugate]] of a [[Definition:Unitary Matrix|unitary matrix]] is [[Definition:Unitary Matrix|unitary]]. | Let $\mathbf U$ be [[Definition:Unitary Matrix|unitary]].
Then:
{{begin-eqn}}
{{eqn | l = \mathbf U
| r = \mathbf U
}}
{{eqn | ll=\leadsto
| l = \paren {\mathbf U^\dagger}^\dagger
| r = \paren {\mathbf U^{-1} }^{-1}
| c = [[Hermitian Conjugate is Involution]]
}}
{{eqn | ll=\leadsto
| l = ... | Hermitian Conjugate of Unitary Matrix is Unitary | https://proofwiki.org/wiki/Hermitian_Conjugate_of_Unitary_Matrix_is_Unitary | https://proofwiki.org/wiki/Hermitian_Conjugate_of_Unitary_Matrix_is_Unitary | [
"Unitary Matrices",
"Hermitian Conjugates"
] | [
"Definition:Hermitian Conjugate",
"Definition:Unitary Matrix",
"Definition:Unitary Matrix"
] | [
"Definition:Unitary Matrix",
"Hermitian Conjugate is Involution",
"Definition:Unitary Matrix",
"Definition:Unitary Matrix",
"Category:Unitary Matrices",
"Category:Hermitian Conjugates"
] |
proofwiki-23510 | Countable Anticompact Space has Countable K-Network | Let $T = \struct {S, \tau}$ be a countable anticompact topological space.
Then $T$ has a countable $k$-network. | {{Recall|K-Network|$k$-network}}
{{:Definition:K-Network/K-Network}}
Let $T = \struct {S, \tau}$ be a countable anticompact topological space.
{{Recall|Anticompact Space|anticompact space}}
{{:Definition:Anticompact Space}}
Let $\VV$ be the set of all singleton subsets of $S$.
For all $U \in \VV$, $U$ is a compact subs... | Let $T = \struct {S, \tau}$ be a [[Definition:Countable Topological Space|countable]] [[Definition:Anticompact Space|anticompact topological space]].
Then $T$ has a [[Definition:Countable Set|countable]] [[Definition:K-Network|$k$-network]]. | {{Recall|K-Network|$k$-network}}
{{:Definition:K-Network/K-Network}}
Let $T = \struct {S, \tau}$ be a [[Definition:Countable Topological Space|countable]] [[Definition:Anticompact Space|anticompact topological space]].
{{Recall|Anticompact Space|anticompact space}}
{{:Definition:Anticompact Space}}
Let $\VV$ be the ... | Countable Anticompact Space has Countable K-Network | https://proofwiki.org/wiki/Countable_Anticompact_Space_has_Countable_K-Network | https://proofwiki.org/wiki/Countable_Anticompact_Space_has_Countable_K-Network | [
"Anticompact Spaces",
"Countable Topological Spaces",
"K-Networks"
] | [
"Definition:Countable Topological Space",
"Definition:Anticompact Space",
"Definition:Countable Set",
"Definition:Network (Topology)/K-Network"
] | [
"Definition:Countable Topological Space",
"Definition:Anticompact Space",
"Definition:Set",
"Definition:Singleton",
"Definition:Subset",
"Definition:Compact Topological Space/Subspace",
"Definition:Compact Topological Space/Subspace",
"Definition:Set Union",
"Definition:Element",
"Definition:Netwo... |
proofwiki-23511 | T3 Space with Sigma-Locally Finite Basis is Pseudometrizable | Let $T = \struct {S, \tau}$ be a $T_3$ topological space.
Let $\BB$ be a $\sigma$-locally finite basis of $T$.
Then $T$ is pseudometrizable. | {{ProofWanted|see Nagata-Smirnov Metrization Theorem/Sufficient Condition}} | Let $T = \struct {S, \tau}$ be a [[Definition:T3 Space|$T_3$ topological space]].
Let $\BB$ be a [[Definition:Sigma-Locally Finite Basis|$\sigma$-locally finite basis]] of $T$.
Then $T$ is [[Definition:Pseudometrizable Space|pseudometrizable]]. | {{ProofWanted|see [[Nagata-Smirnov Metrization Theorem/Sufficient Condition]]}} | T3 Space with Sigma-Locally Finite Basis is Pseudometrizable | https://proofwiki.org/wiki/T3_Space_with_Sigma-Locally_Finite_Basis_is_Pseudometrizable | https://proofwiki.org/wiki/T3_Space_with_Sigma-Locally_Finite_Basis_is_Pseudometrizable | [
"T3 Spaces",
"Sigma-Locally Finite Bases",
"Pseudometrizable Topologies"
] | [
"Definition:T3 Space",
"Definition:Sigma-Locally Finite Basis",
"Definition:Pseudometrizable Topology"
] | [
"Nagata-Smirnov Metrization Theorem/Sufficient Condition"
] |
proofwiki-23512 | Spectral Theorem for Real Symmetric Matrices | Let $\mathbf A$ be a square matrix.
Then $\mathbf A$ is a real symmetric matrix {{Iff}} it is diagonalizable to a real diagonal matrix via an orthogonal transformation.
That is, we can write:
:$\mathbf A = \mathbf U \mathbf D \mathbf U^\intercal$
where:
:$\mathbf D$ is a real diagonal matrix
:$\mathbf U$ is a real orth... | === Necessary Case ===
Let $\mathbf A$ be diagonalizable to a real diagonal matrix via an orthogonal transformation.
Then:
{{begin-eqn}}
{{eqn | l = \mathbf A^\intercal
| r = \paren {\mathbf U \mathbf D \mathbf U^\intercal}^\intercal
}}
{{eqn | r = \mathbf U \mathbf D^\intercal \mathbf U^\intercal
| c = Tra... | Let $\mathbf A$ be a [[Definition:Square Matrix|square matrix]].
Then $\mathbf A$ is a [[Definition:Real Matrix|real]] [[Definition:Symmetric Matrix|symmetric matrix]] {{Iff}} it is [[Definition:Diagonalizable Matrix|diagonalizable]] to a [[Definition:Real Matrix|real]] [[Definition:Diagonal Matrix|diagonal matrix]] v... | === Necessary Case ===
Let $\mathbf A$ be [[Definition:Diagonalizable Matrix|diagonalizable]] to a [[Definition:Real Matrix|real]] [[Definition:Diagonal Matrix|diagonal matrix]] via an [[Definition:Orthogonal Transformation|orthogonal transformation]].
Then:
{{begin-eqn}}
{{eqn | l = \mathbf A^\intercal
| r = \... | Spectral Theorem for Real Symmetric Matrices | https://proofwiki.org/wiki/Spectral_Theorem_for_Real_Symmetric_Matrices | https://proofwiki.org/wiki/Spectral_Theorem_for_Real_Symmetric_Matrices | [
"Spectral Theorems",
"Symmetric Matrices",
"Diagonalizable Matrices",
"Real Matrices"
] | [
"Definition:Matrix/Square Matrix",
"Definition:Real Matrix",
"Definition:Symmetric Matrix",
"Definition:Diagonalizable Matrix",
"Definition:Real Matrix",
"Definition:Diagonal Matrix",
"Definition:Orthogonal Transformation",
"Definition:Real Matrix",
"Definition:Diagonal Matrix",
"Definition:Real M... | [
"Definition:Diagonalizable Matrix",
"Definition:Real Matrix",
"Definition:Diagonal Matrix",
"Definition:Orthogonal Transformation",
"Transpose of Matrix Product",
"Transpose of Transpose of Matrix",
"Definition:Diagonal Matrix",
"Definition:Symmetric Matrix",
"Definition:Real Matrix",
"Definition:... |
proofwiki-23513 | Hermitian Conjugate of Real Matrix is Transpose | Let $\mathbf A$ be a real matrix.
Then:
:$\mathbf A^\dagger = \mathbf A^\intercal$
where
:$\mathbf A^\dagger$ is the Hermitian conjugate of $\mathbf A$
:$\mathbf A^\intercal$ is the transpose of $\mathbf A$ | {{begin-eqn}}
{{eqn | l = \mathbf A^\dagger
| r = \overline {\paren {\mathbf A^\intercal} }
| c = {{Defof|Hermitian Conjugate}}
}}
{{eqn | r = \mathbf A^\intercal
| c = Complex Number equals Conjugate iff Wholly Real
}}
{{end-eqn}}
{{qed}}
Category:Hermitian Conjugates
Category:Transposes of Matrices
... | Let $\mathbf A$ be a [[Definition:Real Matrix|real matrix]].
Then:
:$\mathbf A^\dagger = \mathbf A^\intercal$
where
:$\mathbf A^\dagger$ is the [[Definition:Hermitian Conjugate|Hermitian conjugate]] of $\mathbf A$
:$\mathbf A^\intercal$ is the [[Definition:Transpose of Matrix|transpose]] of $\mathbf A$ | {{begin-eqn}}
{{eqn | l = \mathbf A^\dagger
| r = \overline {\paren {\mathbf A^\intercal} }
| c = {{Defof|Hermitian Conjugate}}
}}
{{eqn | r = \mathbf A^\intercal
| c = [[Complex Number equals Conjugate iff Wholly Real]]
}}
{{end-eqn}}
{{qed}}
[[Category:Hermitian Conjugates]]
[[Category:Transposes ... | Hermitian Conjugate of Real Matrix is Transpose | https://proofwiki.org/wiki/Hermitian_Conjugate_of_Real_Matrix_is_Transpose | https://proofwiki.org/wiki/Hermitian_Conjugate_of_Real_Matrix_is_Transpose | [
"Hermitian Conjugates",
"Transposes of Matrices"
] | [
"Definition:Real Matrix",
"Definition:Hermitian Conjugate",
"Definition:Transpose of Matrix"
] | [
"Complex Number equals Conjugate iff Wholly Real",
"Category:Hermitian Conjugates",
"Category:Transposes of Matrices"
] |
proofwiki-23514 | Diagonal Functor on Product Category has Left Adjoint Iff Category has Binary Coproducts/Necessary Condition | Let $\mathbf C$ be a locally small category.
Let $\mathbf C \times \mathbf C$ be the product category of $\mathbf C$ with itself.
Let $\Delta: \mathbf C \to \mathbf C \times \mathbf C$ denote the diagonal functor.
Let $\Delta$ have a left adjoint.
Then $\mathbf C$ has all binary coproducts
such that the left adjoint is... | Let $\oplus:\mathbf {C \times C} \to \mathbf C$ be a left adjoint of $\Delta$.
==== $\mathbf C$ has all Binary Products ====
By definition of left adjoint there exists an adjunction $\tuple{\oplus, \Delta, \alpha}$.
Let $\iota : \operatorname{id}_{\mathbf {C \times C} } \to \Delta \oplus$ denote the unit of adjunction ... | Let $\mathbf C$ be a [[Definition:Locally Small Category|locally small category]].
Let $\mathbf C \times \mathbf C$ be the [[Definition:Product Category|product category]] of $\mathbf C$ with itself.
Let $\Delta: \mathbf C \to \mathbf C \times \mathbf C$ denote the [[Definition:Diagonal Functor on Product Category|di... | Let $\oplus:\mathbf {C \times C} \to \mathbf C$ be a [[Definition:Left Adjoint Functor|left adjoint]] of $\Delta$.
==== $\mathbf C$ has all Binary Products ====
By definition of [[Definition:Left Adjoint Functor|left adjoint]] there exists an [[Definition:Adjunction|adjunction]] $\tuple{\oplus, \Delta, \alpha}$.
Let... | Diagonal Functor on Product Category has Left Adjoint Iff Category has Binary Coproducts/Necessary Condition | https://proofwiki.org/wiki/Diagonal_Functor_on_Product_Category_has_Left_Adjoint_Iff_Category_has_Binary_Coproducts/Necessary_Condition | https://proofwiki.org/wiki/Diagonal_Functor_on_Product_Category_has_Left_Adjoint_Iff_Category_has_Binary_Coproducts/Necessary_Condition | [
"Diagonal Functor on Product Category has Left Adjoint Iff Category has Binary Coproducts"
] | [
"Definition:Locally Small Category",
"Definition:Product Category",
"Definition:Diagonal Functor/Product Category",
"Definition:Left Adjoint Functor",
"Definition:Coproduct",
"Definition:Left Adjoint Functor",
"Definition:Coproduct Functor"
] | [
"Definition:Left Adjoint Functor",
"Definition:Left Adjoint Functor",
"Definition:Adjunction",
"Definition:Unit of Adjunction",
"Morphism of Unit of Adjunction is Universal",
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Definition:Universal Morphism from Object to Functor",
"Defini... |
proofwiki-23515 | Diagonal Functor on Product Category has Left Adjoint Iff Category has Binary Coproducts/Sufficient Condition | Let $\mathbf C$ be a locally small category.
Let $\mathbf C \times \mathbf C$ be the product category of $\mathbf C$ with itself.
Let $\Delta: \mathbf C \to \mathbf C \times \mathbf C$ denote the diagonal functor.
Let $\mathbf C$ have all binary coproducts.
Then $\Delta$ has a left adjoint. | Let $\sqcup: \mathbf {C \times C} \to \mathbf C$ denote the coproduct functor.
It will be shown that there exists an adjunction $\tuple {\sqcup, \Delta, \alpha}$.
==== Construction of $\alpha$ ====
We construct $\alpha$ as follows.
Let $X \in \mathbf C$ and $\tuple{C_1, C_2} \in \mathbf{C \times C}$ be objects.
Let $\t... | Let $\mathbf C$ be a [[Definition:Locally Small Category|locally small category]].
Let $\mathbf C \times \mathbf C$ be the [[Definition:Product Category|product category]] of $\mathbf C$ with itself.
Let $\Delta: \mathbf C \to \mathbf C \times \mathbf C$ denote the [[Definition:Diagonal Functor on Product Category|di... | Let $\sqcup: \mathbf {C \times C} \to \mathbf C$ denote the [[Definition:Coproduct Functor|coproduct functor]].
It will be shown that there exists an [[Definition:Adjunction|adjunction]] $\tuple {\sqcup, \Delta, \alpha}$.
==== Construction of $\alpha$ ====
We construct $\alpha$ as follows.
Let $X \in \mathbf C$ ... | Diagonal Functor on Product Category has Left Adjoint Iff Category has Binary Coproducts/Sufficient Condition | https://proofwiki.org/wiki/Diagonal_Functor_on_Product_Category_has_Left_Adjoint_Iff_Category_has_Binary_Coproducts/Sufficient_Condition | https://proofwiki.org/wiki/Diagonal_Functor_on_Product_Category_has_Left_Adjoint_Iff_Category_has_Binary_Coproducts/Sufficient_Condition | [
"Diagonal Functor on Product Category has Left Adjoint Iff Category has Binary Coproducts"
] | [
"Definition:Locally Small Category",
"Definition:Product Category",
"Definition:Diagonal Functor/Product Category",
"Definition:Coproduct",
"Definition:Left Adjoint Functor"
] | [
"Definition:Coproduct Functor",
"Definition:Adjunction",
"Definition:Object (Category Theory)",
"Definition:Coproduct",
"Definition:Coproduct/Binary Coproduct Injection",
"Definition:Coproduct",
"Definition:Ordered Tuple as Ordered Set/Ordered Triple",
"Definition:Binary Coproduct UMP",
"Definition:... |
proofwiki-23516 | Identity Morphism of Coproduct | Let $\mathbf C$ be a metacategory.
Let $C$ and $D$ be objects of $\mathbf C$, and let $C \sqcup D$ be a binary coproduct for $C$ and $D$.
Then:
:$\operatorname{id}_{\paren {C \mathop \sqcup D} } = \operatorname{id}_C \sqcup \operatorname{id}_D$
where $\operatorname{id}$ denotes an identity morphism, and $\sqcup$ signif... | By definition of the coproduct morphism $\operatorname{id}_C \sqcup \operatorname{id}_D$, it is the unique morphism making:
$\quad\quad \begin{xy}\xymatrix@+1em@L+5px{
C
\ar[r]^*+{i_1}
\ar[d]_*+{\operatorname{id}_C}
&
C \sqcup D
\ar@{-->}[d]^*+{\operatorname{id}_C \sqcup \operatorname{id}_D}
&
D
\ar[l]_*+{i_2}
\ar[d]^*... | Let $\mathbf C$ be a [[Definition:Metacategory|metacategory]].
Let $C$ and $D$ be [[Definition:Object|objects]] of $\mathbf C$, and let $C \sqcup D$ be a [[Definition:Binary Coproduct (Category Theory)|binary coproduct]] for $C$ and $D$.
Then:
:$\operatorname{id}_{\paren {C \mathop \sqcup D} } = \operatorname{id}_C... | By definition of the [[Definition:Coproduct of Morphisms|coproduct morphism]] $\operatorname{id}_C \sqcup \operatorname{id}_D$, it is the [[Definition:Unique|unique]] [[Definition:Morphism (Category Theory)|morphism]] making:
$\quad\quad \begin{xy}\xymatrix@+1em@L+5px{
C
\ar[r]^*+{i_1}
\ar[d]_*+{\operatorname{id}_C}
&... | Identity Morphism of Coproduct | https://proofwiki.org/wiki/Identity_Morphism_of_Coproduct | https://proofwiki.org/wiki/Identity_Morphism_of_Coproduct | [
"Morphisms"
] | [
"Definition:Metacategory",
"Definition:Object",
"Definition:Coproduct",
"Definition:Identity Morphism",
"Definition:Coproduct of Morphisms"
] | [
"Definition:Coproduct of Morphisms",
"Definition:Unique",
"Definition:Morphism",
"Definition:Commutative Diagram",
"Definition:Identity Morphism",
"Definition:Commutative Diagram",
"Definition:Identity Morphism",
"Definition:Unique",
"Definition:Morphism",
"Category:Morphisms"
] |
proofwiki-23517 | Coproduct of Composite Morphisms | Let $\mathbf C$ be a metacategory.
Let $f \sqcup f': A \sqcup A' \to B \sqcup B'$ and $g \sqcup g': B \sqcup B' \to C \sqcup C'$ be two composable coproducts of morphisms in $\mathbf C$.
Then:
:$\paren {g \circ f} \sqcup \paren {g' \circ f'} = \paren {g \sqcup g'} \circ \paren {f \sqcup f'}$
where $\sqcup$ signifies co... | The situation is efficiently captured in the following commutative diagram:
$\quad\quad \begin{xy}
<-5em,0em>*+{A} = "A",
<0em,0em>*+{A \sqcup A'} = "P",
<5em,0em>*+{A'} = "A2",
<-5em,-5em>*+{B} = "B",
<0em,-5em>*+{B \sqcup B'} = "Q",
<5em,-5em>*+{B'} = "B2",
<-5em,-10em>*+{... | Let $\mathbf C$ be a [[Definition:Metacategory|metacategory]].
Let $f \sqcup f': A \sqcup A' \to B \sqcup B'$ and $g \sqcup g': B \sqcup B' \to C \sqcup C'$ be two [[Definition:Composable Morphisms|composable]] [[Definition:Coproduct of Morphisms|coproducts of morphisms]] in $\mathbf C$.
Then:
:$\paren {g \circ f} ... | The situation is efficiently captured in the following [[Definition:Commutative Diagram|commutative diagram]]:
$\quad\quad \begin{xy}
<-5em,0em>*+{A} = "A",
<0em,0em>*+{A \sqcup A'} = "P",
<5em,0em>*+{A'} = "A2",
<-5em,-5em>*+{B} = "B",
<0em,-5em>*+{B \sqcup B'} = "Q",
<5em,-5em>*+{B... | Coproduct of Composite Morphisms | https://proofwiki.org/wiki/Coproduct_of_Composite_Morphisms | https://proofwiki.org/wiki/Coproduct_of_Composite_Morphisms | [
"Product Categories",
"Morphisms"
] | [
"Definition:Metacategory",
"Definition:Composable Morphisms",
"Definition:Coproduct of Morphisms",
"Definition:Coproduct of Morphisms"
] | [
"Definition:Commutative Diagram",
"Definition:Coproduct of Morphisms",
"Definition:Unique",
"Definition:Morphism",
"Definition:Commutative Diagram",
"Category:Product Categories",
"Category:Morphisms"
] |
proofwiki-23518 | Lindelöf Space is Meta-Lindelöf | Let $T = \struct{S, \tau}$ be a Lindelöf space.
Then $T = \struct{S, \tau}$ is also a meta-Lindelöf space. | {{Recall|Meta-Lindelöf Space|meta-Lindelöf space}}
{{:Definition:Meta-Lindelöf Space}}
Let $T = \struct{S, \tau}$ be a Lindelöf space.
{{Recall|Lindelöf Space|Lindelöf space}}
{{:Definition:Lindelöf Space}}
Let $\CC$ be an open cover of $S$.
Then by Countable Cover is Point Countable, $\CC$ is an open point countable c... | Let $T = \struct{S, \tau}$ be a [[Definition:Lindelöf Space|Lindelöf space]].
Then $T = \struct{S, \tau}$ is also a [[Definition:Meta-Lindelöf Space|meta-Lindelöf space]]. | {{Recall|Meta-Lindelöf Space|meta-Lindelöf space}}
{{:Definition:Meta-Lindelöf Space}}
Let $T = \struct{S, \tau}$ be a [[Definition:Lindelöf Space|Lindelöf space]].
{{Recall|Lindelöf Space|Lindelöf space}}
{{:Definition:Lindelöf Space}}
Let $\CC$ be an [[Definition:Open Cover|open cover]] of $S$.
Then by [[Countabl... | Lindelöf Space is Meta-Lindelöf | https://proofwiki.org/wiki/Lindelöf_Space_is_Meta-Lindelöf | https://proofwiki.org/wiki/Lindelöf_Space_is_Meta-Lindelöf | [
"Meta-Lindelöf Spaces",
"Lindelöf Spaces"
] | [
"Definition:Lindelöf Space",
"Definition:Meta-Lindelöf Space"
] | [
"Definition:Lindelöf Space",
"Definition:Open Cover",
"Countable Cover is Point Countable",
"Definition:Open Cover",
"Definition:Point Countable Cover",
"Cover is Refinement of Itself",
"Definition:Refinement of Cover",
"Definition:Point Countable Cover",
"Definition:Open Refinement"
] |
proofwiki-23519 | Countable Cover is Point Countable | Let $S$ be a set.
Let $\CC$ be a countable cover of $S$.
Then $\CC$ is a point countable cover of $S$. | Let $\CC$ be a countable cover of a set $S$.
Let $x$ be a point of $S$.
As $\CC$ is a countable set of sets, $x$ can be an element of only a countable number of elements of $\CC$.
Hence the result, by definition of point countable cover.
{{qed}} | Let $S$ be a [[Definition:Set|set]].
Let $\CC$ be a [[Definition:Countable Cover|countable cover]] of $S$.
Then $\CC$ is a [[Definition:Point Countable Cover|point countable cover]] of $S$. | Let $\CC$ be a [[Definition:Countable Cover|countable cover]] of a [[Definition:Set|set]] $S$.
Let $x$ be a [[Definition:Point of Set|point]] of $S$.
As $\CC$ is a [[Definition:Countable Set|countable set]] of [[Definition:Set|sets]], $x$ can be an [[Definition:Element|element]] of only a [[Definition:Countable Set|c... | Countable Cover is Point Countable | https://proofwiki.org/wiki/Countable_Cover_is_Point_Countable | https://proofwiki.org/wiki/Countable_Cover_is_Point_Countable | [
"Countable Covers",
"Point Countable Covers"
] | [
"Definition:Set",
"Definition:Cover of Set/Countable",
"Definition:Point Countable Cover"
] | [
"Definition:Cover of Set/Countable",
"Definition:Set",
"Definition:Element",
"Definition:Countable Set",
"Definition:Set",
"Definition:Element",
"Definition:Countable Set",
"Definition:Element",
"Definition:Point Countable Cover"
] |
proofwiki-23520 | Cover is Refinement of Itself | Let $S$ be a set.
Let $\CC$ be a cover for $S$.
Then $\CC$ is a refinement of $\CC$ itself. | From Set is Subset of Itself, $\CC$ is a subcover of itself.
From Subcover is Refinement of Cover, $\CC$ is a refinement of itself.
Hence the result.
{{qed}}
Category:Refinements of Covers
Category:Covers
c3yb90hrnczsaefmu905f6vm5usxiub | Let $S$ be a [[Definition:Set|set]].
Let $\CC$ be a [[Definition:Cover of Set|cover]] for $S$.
Then $\CC$ is a [[Definition:Refinement of Cover|refinement]] of $\CC$ itself. | From [[Set is Subset of Itself]], $\CC$ is a [[Definition:Subcover|subcover]] of itself.
From [[Subcover is Refinement of Cover]], $\CC$ is a [[Definition:Refinement of Cover|refinement]] of itself.
Hence the result.
{{qed}}
[[Category:Refinements of Covers]]
[[Category:Covers]]
c3yb90hrnczsaefmu905f6vm5usxiub | Cover is Refinement of Itself | https://proofwiki.org/wiki/Cover_is_Refinement_of_Itself | https://proofwiki.org/wiki/Cover_is_Refinement_of_Itself | [
"Refinements of Covers",
"Covers"
] | [
"Definition:Set",
"Definition:Cover of Set",
"Definition:Refinement of Cover"
] | [
"Set is Subset of Itself",
"Definition:Subcover",
"Subcover is Refinement of Cover",
"Definition:Refinement of Cover",
"Category:Refinements of Covers",
"Category:Covers"
] |
proofwiki-23521 | Weakly Locally Compact R1 Space is T3.5 | Let $T = \struct {S, \tau}$ be an $R_1$ space which is weakly locally compact.
Then $T$ is a $T_{3 \frac 1 2}$ space. | {{Recall|T3.5 Space|$T_{3 \frac 1 2}$ space}}
{{:Definition:T3.5 Space/Definition 1}}
Let $T = \struct {S, \tau}$ be a weakly locally compact $R_1$ space.
{{Recall|Weakly Locally Compact Space|weakly locally compact space}}
{{:Definition:Weakly Locally Compact Space}}
{{ProofWanted}} | Let $T = \struct {S, \tau}$ be an [[Definition:R1 Space|$R_1$ space]] which is [[Definition:Weakly Locally Compact Space|weakly locally compact]].
Then $T$ is a [[Definition:T3.5 Space|$T_{3 \frac 1 2}$ space]]. | {{Recall|T3.5 Space|$T_{3 \frac 1 2}$ space}}
{{:Definition:T3.5 Space/Definition 1}}
Let $T = \struct {S, \tau}$ be a [[Definition:Weakly Locally Compact Space|weakly locally compact]] [[Definition:R1 Space|$R_1$ space]].
{{Recall|Weakly Locally Compact Space|weakly locally compact space}}
{{:Definition:Weakly Local... | Weakly Locally Compact R1 Space is T3.5 | https://proofwiki.org/wiki/Weakly_Locally_Compact_R1_Space_is_T3.5 | https://proofwiki.org/wiki/Weakly_Locally_Compact_R1_Space_is_T3.5 | [
"Weakly Locally Compact Spaces",
"T3.5 Spaces",
"R1 Spaces",
"Sequence of Implications of Compactness Properties in Hausdorff Space"
] | [
"Definition:R1 Space",
"Definition:Weakly Locally Compact Space",
"Definition:T3.5 Space"
] | [
"Definition:Weakly Locally Compact Space",
"Definition:R1 Space"
] |
proofwiki-23522 | T2 Space is R1 | Let $\struct {S, \tau}$ be a $T_2$ (Hausdorff) space.
Then $\struct {S, \tau}$ is also a $R_1$ space. | Let $T$ be a $T_2$ space.
{{Recall|T2 Space|$T_2$ space}}
{{:Definition:T2 Space/Definition 3}}
By $T_2$ Space is $T_1$ and $T_1$ Space is $T_0$, $T$ is a $T_0$ space.
Hence by definition, for every pair of points $x, y \in T$, $x$ and $y$ are topologically distinguishable.
{{Recall|R1 Space|$R_1$ space}}
{{:Definition... | Let $\struct {S, \tau}$ be a [[Definition:T2 Space|$T_2$ (Hausdorff) space]].
Then $\struct {S, \tau}$ is also a [[Definition:R1 Space|$R_1$ space]]. | Let $T$ be a [[Definition:T2 Space|$T_2$ space]].
{{Recall|T2 Space|$T_2$ space}}
{{:Definition:T2 Space/Definition 3}}
By [[T2 Space is T1|$T_2$ Space is $T_1$]] and [[T1 Space is T0|$T_1$ Space is $T_0$]], $T$ is a [[Definition:T0 Space|$T_0$ space]].
Hence by definition, for every [[Definition:Doubleton|pair]] of... | T2 Space is R1 | https://proofwiki.org/wiki/T2_Space_is_R1 | https://proofwiki.org/wiki/T2_Space_is_R1 | [
"R1 Spaces",
"Hausdorff Spaces"
] | [
"Definition:T2 Space",
"Definition:R1 Space"
] | [
"Definition:T2 Space",
"T2 Space is T1",
"T1 Space is T0",
"Definition:T0 Space",
"Definition:Doubleton",
"Definition:Element",
"Definition:Topologically Distinguishable"
] |
proofwiki-23523 | Diagonal Functor on Product Category has Left Adjoint Iff Category has Binary Coproducts/Lemma 1 | Let $a_1 : X_1 \to Y, a_2 : X_2 \to Y, b: Y \to Z \in \mathbf C$ be morphisms.
::<nowiki>$\begin{xy}\xymatrix@+1em@L+3px{
X_1
\ar[rd]_*+{a_1}
&
&
X_2
\ar[ld]^*+{a_2}
\\
&
Y
\ar[d]_*+{b}
\\
&
Z
}\end{xy}$</nowiki>
Then:
:$\sqbrk{b \circ a_1, b \circ a_2} = b \circ \sqbrk{a_1, a_2}$ | Consider the diagram:
::<nowiki>$\begin{xy}\xymatrix@+1em@L+3px{
X_1
\ar[rr]^*+{i_1}
\ar[rdr]_*+{a_1}
\ar[rdrd]_*+{b \circ a_1}
&&
X_1 \sqcup X_2
\ar@{-->}[d]^*+{\sqbrk{a_1, a_2} }
&&
X_2
\ar[ll]_*+{i_2}
\ar[ldl]^*+{a_2}
\ar[ldld]^*+{b \circ a_2}
\\
&&
Y
\ar[d]^*+{b}
\\
&&
Z
}\end{xy}$</nowiki>
Thi... | Let $a_1 : X_1 \to Y, a_2 : X_2 \to Y, b: Y \to Z \in \mathbf C$ be [[Definition:Morphism (Category Theory)|morphisms]].
::<nowiki>$\begin{xy}\xymatrix@+1em@L+3px{
X_1
\ar[rd]_*+{a_1}
&
&
X_2
\ar[ld]^*+{a_2}
\\
&
Y
\ar[d]_*+{b}
\\
&
Z
}\end{xy}$</nowiki>
Then:
:$\sqbrk{b \circ a_1, b \circ a_2} = b \circ... | Consider the diagram:
::<nowiki>$\begin{xy}\xymatrix@+1em@L+3px{
X_1
\ar[rr]^*+{i_1}
\ar[rdr]_*+{a_1}
\ar[rdrd]_*+{b \circ a_1}
&&
X_1 \sqcup X_2
\ar@{-->}[d]^*+{\sqbrk{a_1, a_2} }
&&
X_2
\ar[ll]_*+{i_2}
\ar[ldl]^*+{a_2}
\ar[ldld]^*+{b \circ a_2}
\\
&&
Y
\ar[d]^*+{b}
\\
&&
Z
}\end{xy}$</nowiki>
... | Diagonal Functor on Product Category has Left Adjoint Iff Category has Binary Coproducts/Lemma 1 | https://proofwiki.org/wiki/Diagonal_Functor_on_Product_Category_has_Left_Adjoint_Iff_Category_has_Binary_Coproducts/Lemma_1 | https://proofwiki.org/wiki/Diagonal_Functor_on_Product_Category_has_Left_Adjoint_Iff_Category_has_Binary_Coproducts/Lemma_1 | [
"Diagonal Functor on Product Category has Left Adjoint Iff Category has Binary Coproducts"
] | [
"Definition:Morphism"
] | [
"Definition:Commutative Diagram",
"Definition:Binary Coproduct UMP",
"Definition:Binary Coproduct UMP",
"Definition:Commutative Diagram",
"definition:Unique",
"Definition:Morphism",
"Category:Diagonal Functor on Product Category has Left Adjoint Iff Category has Binary Coproducts"
] |
proofwiki-23524 | Diagonal Functor on Product Category has Left Adjoint Iff Category has Binary Coproducts/Lemma 2 | Let $a_1: X_1 \to Y_1, a_2: X_1 \to Y_2, b_1 : Y_1 \to Z, b_2 : Y_2 \to Z \in \mathbf C$ be morphisms:
::<nowiki>$\begin{xy}\xymatrix@+1em@L+3px{
X_1
\ar[d]_*+{a_1}
&&
X_2
\ar[d]^*+{a_2}
\\
Y_1
\ar[rd]_*+{b_1}
&&
Y_2
\ar[ld]^*+{b_2}
\\
&
Z
}\end{xy}$</nowiki>
Then:
:$\sqbrk{b_1 \circ a_1, b_2 \circ a_2} = ... | Consider the diagram:
::<nowiki>$\begin{xy}
<-6em,5em>*+{X_1} = "TL",
<0em,5em>*+{X_1 \sqcup X_2} = "TM",
<6em,5em>*+{X_2} = "TR",
<-6em,0em>*+{Y_1} = "ML",
<0em,0em>*+{Y_1 \sqcup Y_2} = "MM",
<6em,0em>*+{Y_2} = "MR",
<0em,-5em>*+{Z} = "BM",
"TL";"TM" **@{-} ?>*@{>} ?<>(.5)*!/_.8em/{i_1},
"TR";"TM" **@{-} ?>*@{>} ?<>(.... | Let $a_1: X_1 \to Y_1, a_2: X_1 \to Y_2, b_1 : Y_1 \to Z, b_2 : Y_2 \to Z \in \mathbf C$ be [[Definition:Morphism (Category Theory)|morphisms]]:
::<nowiki>$\begin{xy}\xymatrix@+1em@L+3px{
X_1
\ar[d]_*+{a_1}
&&
X_2
\ar[d]^*+{a_2}
\\
Y_1
\ar[rd]_*+{b_1}
&&
Y_2
\ar[ld]^*+{b_2}
\\
&
Z
}\end{xy}$</nowiki>
T... | Consider the diagram:
::<nowiki>$\begin{xy}
<-6em,5em>*+{X_1} = "TL",
<0em,5em>*+{X_1 \sqcup X_2} = "TM",
<6em,5em>*+{X_2} = "TR",
<-6em,0em>*+{Y_1} = "ML",
<0em,0em>*+{Y_1 \sqcup Y_2} = "MM",
<6em,0em>*+{Y_2} = "MR",
<0em,-5em>*+{Z} = "BM",
"TL";"TM" **@{-} ?>*@{>} ?<>(.5)*!/_.8em/{i_1},
"TR";"TM" **@{-} ?>*@{>} ?<... | Diagonal Functor on Product Category has Left Adjoint Iff Category has Binary Coproducts/Lemma 2 | https://proofwiki.org/wiki/Diagonal_Functor_on_Product_Category_has_Left_Adjoint_Iff_Category_has_Binary_Coproducts/Lemma_2 | https://proofwiki.org/wiki/Diagonal_Functor_on_Product_Category_has_Left_Adjoint_Iff_Category_has_Binary_Coproducts/Lemma_2 | [
"Diagonal Functor on Product Category has Right Adjoint Iff Category has Binary Products"
] | [
"Definition:Morphism"
] | [
"Definition:Commutative Diagram",
"Definition:Coproduct of Morphisms",
"Definition:Binary Coproduct UMP",
"Definition:Binary Coproduct UMP",
"Definition:Commutative Diagram",
"definition:Unique",
"Definition:Morphism",
"Category:Diagonal Functor on Product Category has Right Adjoint Iff Category has B... |
proofwiki-23525 | First-Countable T2 Countably Compact Space is Regular | Let $\struct {S, \tau}$ be a first-countable $T_2$ (Hausdorff) space which is countably compact.
Then $\struct {S, \tau}$ is also a regular space. | {{Recall|Regular Space|regular space}}
{{:Definition:Regular Space/Definition 3}}
So, let $\struct {S, \tau}$ be a first-countable $T_2$ (Hausdorff) space which is countably compact.
Let $A$ be a closed set of $T$.
Let $x \notin A$ be a point of $S$ which is not in $A$.
{{Recall|First-Countable Space|first-countable sp... | Let $\struct {S, \tau}$ be a [[Definition:First-Countable Space|first-countable]] [[Definition:T2 Space|$T_2$ (Hausdorff) space]] which is [[Definition:Countably Compact Space|countably compact]].
Then $\struct {S, \tau}$ is also a [[Definition:Regular Space|regular space]]. | {{Recall|Regular Space|regular space}}
{{:Definition:Regular Space/Definition 3}}
So, let $\struct {S, \tau}$ be a [[Definition:First-Countable Space|first-countable]] [[Definition:T2 Space|$T_2$ (Hausdorff) space]] which is [[Definition:Countably Compact Space|countably compact]].
Let $A$ be a [[Definition:Closed Se... | First-Countable T2 Countably Compact Space is Regular | https://proofwiki.org/wiki/First-Countable_T2_Countably_Compact_Space_is_Regular | https://proofwiki.org/wiki/First-Countable_T2_Countably_Compact_Space_is_Regular | [
"First-Countable Spaces",
"Hausdorff Spaces",
"Countably Compact Spaces",
"Regular Spaces"
] | [
"Definition:First-Countable Space",
"Definition:T2 Space",
"Definition:Countably Compact Space",
"Definition:Regular Space"
] | [
"Definition:First-Countable Space",
"Definition:T2 Space",
"Definition:Countably Compact Space",
"Definition:Closed Set/Topology",
"Definition:Element",
"Definition:Countable Set",
"Definition:Local Basis",
"Definition:T2 Space",
"Definition:Countable Set",
"Definition:Open Set/Topology",
"Defin... |
proofwiki-23526 | Topological Space with Countable Network has Sigma-Locally Finite Network | Let $T = \struct {S, \tau}$ be a topological space which has a countable network.
Then $T$ has a $\sigma$-locally finite network. | {{ProofWanted|Clarification needed as to the nature of a $\sigma$-locally finite set decoupled from the context of a topological space}} | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]] which has a [[Definition:Countable Set|countable]] [[Definition:Network (Topology)|network]].
Then $T$ has a [[Definition:Sigma-Locally Finite Set of Subsets|$\sigma$-locally finite]] [[Definition:Network (Topology)|network]]. | {{ProofWanted|Clarification needed as to the nature of a [[Definition:Sigma-Locally Finite Set of Subsets|$\sigma$-locally finite set]] decoupled from the context of a topological space}} | Topological Space with Countable Network has Sigma-Locally Finite Network | https://proofwiki.org/wiki/Topological_Space_with_Countable_Network_has_Sigma-Locally_Finite_Network | https://proofwiki.org/wiki/Topological_Space_with_Countable_Network_has_Sigma-Locally_Finite_Network | [
"Networks (Topology)",
"Countable Sets",
"Sigma-Locally Finite Sets of Subsets"
] | [
"Definition:Topological Space",
"Definition:Countable Set",
"Definition:Network (Topology)",
"Definition:Sigma-Locally Finite Set of Subsets",
"Definition:Network (Topology)"
] | [
"Definition:Sigma-Locally Finite Set of Subsets"
] |
proofwiki-23527 | Strongly Paracompact Space is Paracompact | Let $T = \struct {S, \tau}$ be a strongly paracompact space.
Then $T$ is paracompact. | {{Recall|Paracompact Space|paracompact space}}
{{:Definition:Paracompact Space}}
Let $T = \struct {S, \tau}$ be a strongly paracompact space.
{{Recall|Strongly Paracompact Space|strongly paracompact space}}
{{:Definition:Strongly Paracompact Space}}
The result follows directly from Star-Finite Set is Locally Finite.
{{... | Let $T = \struct {S, \tau}$ be a [[Definition:Strongly Paracompact Space|strongly paracompact space]].
Then $T$ is [[Definition:Paracompact Space|paracompact]]. | {{Recall|Paracompact Space|paracompact space}}
{{:Definition:Paracompact Space}}
Let $T = \struct {S, \tau}$ be a [[Definition:Strongly Paracompact Space|strongly paracompact space]].
{{Recall|Strongly Paracompact Space|strongly paracompact space}}
{{:Definition:Strongly Paracompact Space}}
The result follows direct... | Strongly Paracompact Space is Paracompact | https://proofwiki.org/wiki/Strongly_Paracompact_Space_is_Paracompact | https://proofwiki.org/wiki/Strongly_Paracompact_Space_is_Paracompact | [
"Strongly Paracompact Spaces",
"Paracompact Spaces",
"Sequence of Implications of Paracompactness Properties"
] | [
"Definition:Strongly Paracompact Space",
"Definition:Paracompact Space"
] | [
"Definition:Strongly Paracompact Space",
"Star-Finite Set is Locally Finite"
] |
proofwiki-23528 | Star-Finite Set is Locally Finite | Let $S$ be a set.
Let $\FF$ be a set of subsets of $S$ which is '''star-finite'''.
Then $\FF$ is locally finite. | {{Recall|Locally Finite Set of Subsets|locally finite}}
{{:Definition:Locally Finite Set of Subsets}}
Let $\FF \subseteq \powerset S$ be '''star-finite'''.
{{Recall|Star-Finite|star-finite}}
{{:Definition:Star-Finite}}
{{ProofWanted}} | Let $S$ be a [[Definition:Set|set]].
Let $\FF$ be a [[Definition:Set|set]] of [[Definition:Subset|subsets]] of $S$ which is '''[[Definition:Star-Finite|star-finite]]'''.
Then $\FF$ is [[Definition:Locally Finite|locally finite]]. | {{Recall|Locally Finite Set of Subsets|locally finite}}
{{:Definition:Locally Finite Set of Subsets}}
Let $\FF \subseteq \powerset S$ be '''[[Definition:Star-Finite|star-finite]]'''.
{{Recall|Star-Finite|star-finite}}
{{:Definition:Star-Finite}}
{{ProofWanted}} | Star-Finite Set is Locally Finite | https://proofwiki.org/wiki/Star-Finite_Set_is_Locally_Finite | https://proofwiki.org/wiki/Star-Finite_Set_is_Locally_Finite | [
"Star-Finite",
"Locally Finite"
] | [
"Definition:Set",
"Definition:Set",
"Definition:Subset",
"Definition:Star-Finite",
"Definition:Locally Finite"
] | [
"Definition:Star-Finite"
] |
proofwiki-23529 | Topological Space with Sigma-Locally Finite K-Network has Sigma-Locally Finite Network | Let $T = \struct {S, \tau}$ be a topological space which has a $\sigma$-locally finite $k$-network.
Then $T$ has a $\sigma$-locally finite network. | Follows directly from K-Network is Network.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]] which has a [[Definition:Sigma-Locally Finite Set of Subsets|$\sigma$-locally finite]] [[Definition:K-Network|$k$-network]].
Then $T$ has a [[Definition:Sigma-Locally Finite Set of Subsets|$\sigma$-locally finite]] [[Definition:Network... | Follows directly from [[K-Network is Network]].
{{qed}} | Topological Space with Sigma-Locally Finite K-Network has Sigma-Locally Finite Network | https://proofwiki.org/wiki/Topological_Space_with_Sigma-Locally_Finite_K-Network_has_Sigma-Locally_Finite_Network | https://proofwiki.org/wiki/Topological_Space_with_Sigma-Locally_Finite_K-Network_has_Sigma-Locally_Finite_Network | [
"K-Networks",
"Networks (Topology)",
"Sigma-Locally Finite Sets of Subsets"
] | [
"Definition:Topological Space",
"Definition:Sigma-Locally Finite Set of Subsets",
"Definition:Network (Topology)/K-Network",
"Definition:Sigma-Locally Finite Set of Subsets",
"Definition:Network (Topology)"
] | [
"K-Network is Network"
] |
proofwiki-23530 | Taylor Series of Lorentz Factor at Low Speeds | The Lorentz factor has the following Taylor polynomial around $v = 0$:
{{begin-eqn}}
{{eqn | l = \map \gamma v
| r = \frac 1 {\sqrt {1 - \frac {v^2} {c^2} } }
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \frac 1 {n!} \paren {\frac {\paren {2 n - 1} !!} {2 n} } \paren {\frac v c }^{2 n}
}}
{{eqn | o = \approx
... | {{Recall|Lorentz Factor|Lorentz factor}}
{{:Definition:Lorentz Factor}}
Let $x = v^2 / c^2$.
Then:
{{begin-eqn}}
{{eqn | l = \map \gamma x
| r = \frac 1 {\sqrt {1 - x} }
}}
{{eqn | r = \paren {1 - x}^{-\frac 1 2}
}}
{{end-eqn}}
Note that $\map \gamma 0 = 1$.
Proof by induction:
For all $n \in \mathbb N_{> 0}$, le... | The [[Definition:Lorentz Factor|Lorentz factor]] has the following [[Taylor's Theorem/One Variable|Taylor polynomial]] around $v = 0$:
{{begin-eqn}}
{{eqn | l = \map \gamma v
| r = \frac 1 {\sqrt {1 - \frac {v^2} {c^2} } }
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \frac 1 {n!} \paren {\frac {\paren {2 n - 1} !!}... | {{Recall|Lorentz Factor|Lorentz factor}}
{{:Definition:Lorentz Factor}}
Let $x = v^2 / c^2$.
Then:
{{begin-eqn}}
{{eqn | l = \map \gamma x
| r = \frac 1 {\sqrt {1 - x} }
}}
{{eqn | r = \paren {1 - x}^{-\frac 1 2}
}}
{{end-eqn}}
Note that $\map \gamma 0 = 1$.
Proof by [[Definition:Principle of Mathematical I... | Taylor Series of Lorentz Factor at Low Speeds | https://proofwiki.org/wiki/Taylor_Series_of_Lorentz_Factor_at_Low_Speeds | https://proofwiki.org/wiki/Taylor_Series_of_Lorentz_Factor_at_Low_Speeds | [
"Relativistic Mechanics",
"Lorentz Factor",
"Proofs by Induction"
] | [
"Definition:Lorentz Factor",
"Taylor's Theorem/One Variable",
"Definition:Speed",
"Definition:Speed of Light",
"Definition:Double Factorial"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Definition:Derivative/Real Function",
"Definition:Proposition",
"Definition:Derivative/Real Function",
"Definition:Derivative/Real Function",
"Definition:Derivative/Real Function"
] |
proofwiki-23531 | T4 and R0 Space is T3.5 | Let $T = \struct {S, \tau}$ be a $T_4$ space which is also an $R_0$ space.
Then $T$ is also a $T_{3 \frac 1 2}$ space. | {{Recall|T3.5 Space|$T_{3 \frac 1 2}$ space}}
{{:Definition:T3.5 Space/Definition 1}}
Let $T = \struct {S, \tau}$ be a $T_4$ space and also an $R_0$ space
{{Recall|T4 Space|$T_4$ space}}
{{:Definition:T4 Space/Definition 1}}
{{Recall|R0 Space|$R_0$ space}}
{{:Definition:R0 Space}}
{{ProofWanted|Should be similar to Nor... | Let $T = \struct {S, \tau}$ be a [[Definition:T4 Space|$T_4$ space]] which is also an [[Definition:R0 Space|$R_0$ space]].
Then $T$ is also a [[Definition:T3.5 Space|$T_{3 \frac 1 2}$ space]]. | {{Recall|T3.5 Space|$T_{3 \frac 1 2}$ space}}
{{:Definition:T3.5 Space/Definition 1}}
Let $T = \struct {S, \tau}$ be a [[Definition:T4 Space|$T_4$ space]] and also an [[Definition:R0 Space|$R_0$ space]]
{{Recall|T4 Space|$T_4$ space}}
{{:Definition:T4 Space/Definition 1}}
{{Recall|R0 Space|$R_0$ space}}
{{:Definiti... | T4 and R0 Space is T3.5 | https://proofwiki.org/wiki/T4_and_R0_Space_is_T3.5 | https://proofwiki.org/wiki/T4_and_R0_Space_is_T3.5 | [
"T4 Spaces",
"R0 Spaces",
"T3.5 Spaces"
] | [
"Definition:T4 Space",
"Definition:R0 Space",
"Definition:T3.5 Space"
] | [
"Definition:T4 Space",
"Definition:R0 Space",
"Normal Space is T3.5"
] |
proofwiki-23532 | Topological Space with Dispersion Point is not Empty | Let $T = \struct {S, \tau}$ be a topological space which has a dispersion point.
Then $T$ is not the empty space. | Let $T = \struct {S, \tau}$ have a dispersion point $p$.
Then $p \in S$ and so $S \ne \O$.
The result follows.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]] which has a [[Definition:Dispersion Point|dispersion point]].
Then $T$ is not the [[Definition:Empty Space|empty space]]. | Let $T = \struct {S, \tau}$ have a [[Definition:Dispersion Point|dispersion point]] $p$.
Then $p \in S$ and so $S \ne \O$.
The result follows.
{{qed}} | Topological Space with Dispersion Point is not Empty | https://proofwiki.org/wiki/Topological_Space_with_Dispersion_Point_is_not_Empty | https://proofwiki.org/wiki/Topological_Space_with_Dispersion_Point_is_not_Empty | [
"Dispersion Points",
"Empty Topological Space"
] | [
"Definition:Topological Space",
"Definition:Dispersion Point",
"Definition:Empty Topological Space"
] | [
"Definition:Dispersion Point"
] |
proofwiki-23533 | Partition Space is Extremally Disconnected | Let $T = \struct {S, \tau}$ be a partition space.
Then $T$ is extremally disconnected. | {{Recall|Extremally Disconnected Space|extremally disconnected space}}
{{:Definition:Extremally Disconnected Space/Definition 1}}
Let $U \in \tau$ be an open set of $T$.
By Subset of Partition Space is Open iff Closed, $U$ is also closed in $T$.
From Closed Set equals its Closure, $U$ equals its closure.
Hence the resu... | Let $T = \struct {S, \tau}$ be a [[Definition:Partition Space|partition space]].
Then $T$ is [[Definition:Extremally Disconnected Space|extremally disconnected]]. | {{Recall|Extremally Disconnected Space|extremally disconnected space}}
{{:Definition:Extremally Disconnected Space/Definition 1}}
Let $U \in \tau$ be an [[Definition:Open Set (Topology)|open set]] of $T$.
By [[Subset of Partition Space is Open iff Closed]], $U$ is also [[Definition:Closed Set (Topology)|closed]] in $... | Partition Space is Extremally Disconnected | https://proofwiki.org/wiki/Partition_Space_is_Extremally_Disconnected | https://proofwiki.org/wiki/Partition_Space_is_Extremally_Disconnected | [
"Partition Topologies",
"Examples of Extremally Disconnected Spaces"
] | [
"Definition:Partition Topology",
"Definition:Extremally Disconnected Space"
] | [
"Definition:Open Set/Topology",
"Subset of Partition Space is Open iff Closed",
"Definition:Closed Set/Topology",
"Set is Closed iff Equals Topological Closure",
"Definition:Closure (Topology)",
"Definition:Extremally Disconnected Space"
] |
proofwiki-23534 | Totally Pathwise Disconnected Space is T1 | Let $T = \struct {S, \tau}$ be a topological space which is totally pathwise disconnected space.
Then $T$ is a $T_1$ space. | Let $T = \struct {S, \tau}$ be a totally pathwise disconnected space.
By Total Pathwise Disconnectedness is Hereditary, every subspace of $T$ is also totally pathwise disconnected.
In particular, every doubleton subspace of $T$ is totally pathwise disconnected.
Let $a, b \in S$ be arbitrary distinct points.
From Double... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]] which is [[Definition:Totally Pathwise Disconnected Space|totally pathwise disconnected space]].
Then $T$ is a [[Definition:T1 Space|$T_1$ space]]. | Let $T = \struct {S, \tau}$ be a [[Definition:Totally Pathwise Disconnected Space|totally pathwise disconnected space]].
By [[Total Pathwise Disconnectedness is Hereditary]], every [[Definition:Topological Subspace|subspace]] of $T$ is also [[Definition:Totally Pathwise Disconnected Space|totally pathwise disconnected... | Totally Pathwise Disconnected Space is T1/Proof 2 | https://proofwiki.org/wiki/Totally_Pathwise_Disconnected_Space_is_T1 | https://proofwiki.org/wiki/Totally_Pathwise_Disconnected_Space_is_T1/Proof_2 | [
"Totally Pathwise Disconnected Space is T1",
"Totally Pathwise Disconnected Spaces",
"T1 Spaces"
] | [
"Definition:Topological Space",
"Definition:Totally Pathwise Disconnected Space",
"Definition:T1 Space"
] | [
"Definition:Totally Pathwise Disconnected Space",
"Total Pathwise Disconnectedness is Hereditary",
"Definition:Topological Subspace",
"Definition:Totally Pathwise Disconnected Space",
"Definition:Doubleton",
"Definition:Topological Subspace",
"Definition:Totally Pathwise Disconnected Space",
"Definiti... |
proofwiki-23535 | Total Pathwise Disconnectedness is Hereditary | Let $T = \struct {S, \tau}$ be a topological space which is totally pathwise disconnected space.
Let $H$ be a subspace of $T$.
Then $H$ is also a totally pathwise disconnected space. | {{Recall|Totally Pathwise Disconnected Space|totally pathwise disconnected space}}
{{:Definition:Totally Pathwise Disconnected Space/Definition 1}}
Let $T = \struct {S, \tau}$ be totally pathwise disconnected.
{{AimForCont}} $H$ is a subspace of $T$ which is not totally pathwise disconnected.
Then by definition there e... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]] which is [[Definition:Totally Pathwise Disconnected Space|totally pathwise disconnected space]].
Let $H$ be a [[Definition:Topological Subspace|subspace]] of $T$.
Then $H$ is also a [[Definition:Totally Pathwise Disconnected Space|to... | {{Recall|Totally Pathwise Disconnected Space|totally pathwise disconnected space}}
{{:Definition:Totally Pathwise Disconnected Space/Definition 1}}
Let $T = \struct {S, \tau}$ be [[Definition:Totally Pathwise Disconnected Space|totally pathwise disconnected]].
{{AimForCont}} $H$ is a [[Definition:Topological Subspace... | Total Pathwise Disconnectedness is Hereditary | https://proofwiki.org/wiki/Total_Pathwise_Disconnectedness_is_Hereditary | https://proofwiki.org/wiki/Total_Pathwise_Disconnectedness_is_Hereditary | [
"Totally Pathwise Disconnected Spaces",
"Examples of Hereditary Properties"
] | [
"Definition:Topological Space",
"Definition:Totally Pathwise Disconnected Space",
"Definition:Topological Subspace",
"Definition:Totally Pathwise Disconnected Space"
] | [
"Definition:Totally Pathwise Disconnected Space",
"Definition:Topological Subspace",
"Definition:Totally Pathwise Disconnected Space",
"Definition:Path Component",
"Definition:Element",
"Definition:Path Component",
"Definition:Totally Pathwise Disconnected Space",
"Definition:Path Component",
"Defin... |
proofwiki-23536 | Doubleton Totally Pathwise Disconnected Space is Discrete | Let $T = \struct {S, \tau}$ be a topological space whose underlying set $S$ is a doubleton.
Let $T$ be totally pathwise disconnected space.
Then $T$ is a discrete space. | {{WLOG}}, let $S = \set {0, 1}$.
Hence let $T = \struct {\set {0, 1}, \tau}$ be a totally pathwise disconnected space.
{{AimForCont}} $T$ is not a discrete space.
{{WLOG}}, let $\set 1$ not be an open set of $T$.
Let $f: \closedint 0 1 \to S$ be the mapping defined as:
:$\map f x = \begin {cases} 0 & : x < \dfrac 1 2 \... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]] whose [[Definition:Underlying Set of Topological Space|underlying set]] $S$ is a [[Definition:Doubleton|doubleton]].
Let $T$ be [[Definition:Totally Pathwise Disconnected Space|totally pathwise disconnected space]].
Then $T$ is a [[D... | {{WLOG}}, let $S = \set {0, 1}$.
Hence let $T = \struct {\set {0, 1}, \tau}$ be a [[Definition:Totally Pathwise Disconnected Space|totally pathwise disconnected space]].
{{AimForCont}} $T$ is not a [[Definition:Discrete Space|discrete space]].
{{WLOG}}, let $\set 1$ not be an [[Definition:Open Set (Topology)|open se... | Doubleton Totally Pathwise Disconnected Space is Discrete | https://proofwiki.org/wiki/Doubleton_Totally_Pathwise_Disconnected_Space_is_Discrete | https://proofwiki.org/wiki/Doubleton_Totally_Pathwise_Disconnected_Space_is_Discrete | [
"Totally Pathwise Disconnected Spaces",
"Discrete Topologies"
] | [
"Definition:Topological Space",
"Definition:Underlying Set/Topological Space",
"Definition:Doubleton",
"Definition:Totally Pathwise Disconnected Space",
"Definition:Discrete Topology"
] | [
"Definition:Totally Pathwise Disconnected Space",
"Definition:Discrete Topology",
"Definition:Open Set/Topology",
"Definition:Mapping",
"Definition:Continuous Mapping (Topology)",
"Definition:Constant Mapping",
"Definition:Contradiction",
"Definition:Totally Pathwise Disconnected Space",
"Proof by C... |
proofwiki-23537 | Pseudometrizable Space is Fully T4 | A pseudometrizable space $T = \struct {S, \tau}$ is a fully $T_4$ space. | {{Recall|Fully T4 Space|fully $T_4$ space}}
{{:Definition:Fully T4 Space}}
{{proof wanted}} | A [[Definition:Pseudometrizable Space|pseudometrizable space]] $T = \struct {S, \tau}$ is a [[Definition:Fully T4 Space|fully $T_4$ space]]. | {{Recall|Fully T4 Space|fully $T_4$ space}}
{{:Definition:Fully T4 Space}}
{{proof wanted}} | Pseudometrizable Space is Fully T4 | https://proofwiki.org/wiki/Pseudometrizable_Space_is_Fully_T4 | https://proofwiki.org/wiki/Pseudometrizable_Space_is_Fully_T4 | [
"Pseudometrizable Topologies",
"Fully T4 Spaces"
] | [
"Definition:Pseudometrizable Topology",
"Definition:Fully T4 Space"
] | [] |
proofwiki-23538 | Locally Pseudometrizable Space is First-Countable | Let $T = \struct {S, \tau}$ be a topological space which is locally pseudometrizable.
Then $T$ is first-countable. | Let $\NN$ be a pseudometrizable neighborhood.
The open balls of rational radius within $\NN$ form a local basis.
Hence the result from the definition of first-countable.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]] which is [[Definition:Locally Pseudometrizable Space|locally pseudometrizable]].
Then $T$ is [[Definition:First-Countable Space|first-countable]]. | Let $\NN$ be a [[Definition:Pseudometrizable Space|pseudometrizable]] [[Definition:Neighborhood of Point|neighborhood]].
The [[Definition:Open Ball|open balls]] of [[Definition:Rational Number|rational]] [[Definition:Radius of Open Ball|radius]] within $\NN$ form a [[Definition:Local Basis|local basis]].
Hence the re... | Locally Pseudometrizable Space is First-Countable | https://proofwiki.org/wiki/Locally_Pseudometrizable_Space_is_First-Countable | https://proofwiki.org/wiki/Locally_Pseudometrizable_Space_is_First-Countable | [
"Locally Pseudometrizable Spaces",
"First-Countable Spaces"
] | [
"Definition:Topological Space",
"Definition:Locally Pseudometrizable Space",
"Definition:First-Countable Space"
] | [
"Definition:Pseudometrizable Topology",
"Definition:Neighborhood (Topology)/Point",
"Definition:Open Ball",
"Definition:Rational Number",
"Definition:Open Ball/Radius",
"Definition:Local Basis",
"Definition:First-Countable Space"
] |
proofwiki-23539 | K1-Space which is K1-Hausdorff is K3 | Let $T = \struct {S, \tau}$ be a topological space which is both a $k_1$-space and a $k_1$-Hausdorff space.
Then $T$ is a $k_3$-space. | {{Recall|K3-Space|$k_3$-space}}
{{:Definition:K3-Space}}
Let $T = \struct {S, \tau}$ be both a $k_1$-space and a $k_1$-Hausdorff space.
{{Recall|K1-Space|$k_1$-space}}
{{:Definition:K1-Space/Definition 1}}
{{Recall|K1-Hausdorff Space|$k_1$-Hausdorff space}}
{{:Definition:K1-Hausdorff Space/Definition 2}}
The result fol... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]] which is both a [[Definition:K1-Space|$k_1$-space]] and a [[Definition:K1-Hausdorff Space|$k_1$-Hausdorff space]].
Then $T$ is a [[Definition:K3-Space|$k_3$-space]]. | {{Recall|K3-Space|$k_3$-space}}
{{:Definition:K3-Space}}
Let $T = \struct {S, \tau}$ be both a [[Definition:K1-Space|$k_1$-space]] and a [[Definition:K1-Hausdorff Space|$k_1$-Hausdorff space]].
{{Recall|K1-Space|$k_1$-space}}
{{:Definition:K1-Space/Definition 1}}
{{Recall|K1-Hausdorff Space|$k_1$-Hausdorff space}}
{... | K1-Space which is K1-Hausdorff is K3 | https://proofwiki.org/wiki/K1-Space_which_is_K1-Hausdorff_is_K3 | https://proofwiki.org/wiki/K1-Space_which_is_K1-Hausdorff_is_K3 | [
"K1-Spaces",
"K1-Hausdorff Spaces",
"K3-Spaces"
] | [
"Definition:Topological Space",
"Definition:K1-Space",
"Definition:K1-Hausdorff Space",
"Definition:K3-Space"
] | [
"Definition:K1-Space",
"Definition:K1-Hausdorff Space"
] |
proofwiki-23540 | Weakly Lindelöf Space with Sigma-Locally Finite Basis is Second-Countable | Let $T = \struct {S, \tau}$ be a weakly Lindelöf topological space.
Let $\BB$ be a $\sigma$-locally finite basis of $T$.
Then $T$ is second-countable. | {{Refactor}} | Let $T = \struct {S, \tau}$ be a [[Definition:Weakly Lindelöf Space|weakly Lindelöf topological space]].
Let $\BB$ be a [[Definition:Sigma-Locally Finite Basis|$\sigma$-locally finite basis]] of $T$.
Then $T$ is [[Definition:Second-Countable Space|second-countable]]. | {{Refactor}} | Weakly Lindelöf Space with Sigma-Locally Finite Basis is Second-Countable | https://proofwiki.org/wiki/Weakly_Lindelöf_Space_with_Sigma-Locally_Finite_Basis_is_Second-Countable | https://proofwiki.org/wiki/Weakly_Lindelöf_Space_with_Sigma-Locally_Finite_Basis_is_Second-Countable | [
"Weakly Lindelöf Spaces",
"Sigma-Locally Finite Bases",
"Second-Countable Spaces"
] | [
"Definition:Weakly Lindelöf Space",
"Definition:Sigma-Locally Finite Basis",
"Definition:Second-Countable Space"
] | [] |
proofwiki-23541 | Point in T1 Space is Intersection of Open Sets | Let $T = \struct {S, \tau}$ be a topological space.
Then:
:$T$ is a $T_1$ space
{{iff}}
:for every point $x \in S$, the singleton $\set x$ is the intersection of every open set $U$ of $T$ such that $x \in U$:
::$\forall x \in S: \set x = \ds \bigcap_{x \mathop \in U} \set {U \in \tau}$ | {{Recall|T1 Space|$T_1$ space}}
{{:Definition:T1 Space/Definition 1}}
For arbitrary $x \in S$, let $\CC_x$ be the intersection of every open set $U$ of $T$ such that $x \in U$:
:$\CC_x = \ds \bigcap_{x \mathop \in U} \set {U \in \tau}$ | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Then:
:$T$ is a [[Definition:T1 Space|$T_1$ space]]
{{iff}}
:for every [[Definition:Point of Set|point]] $x \in S$, the [[Definition:Singleton|singleton]] $\set x$ is the [[Definition:Set Intersection|intersection]] of every [[Definit... | {{Recall|T1 Space|$T_1$ space}}
{{:Definition:T1 Space/Definition 1}}
For [[Definition:Arbitrary|arbitrary]] $x \in S$, let $\CC_x$ be the [[Definition:Set Intersection|intersection]] of every [[Definition:Open Set (Topology)|open set]] $U$ of $T$ such that $x \in U$:
:$\CC_x = \ds \bigcap_{x \mathop \in U} \set {U \i... | Point in T1 Space is Intersection of Open Sets | https://proofwiki.org/wiki/Point_in_T1_Space_is_Intersection_of_Open_Sets | https://proofwiki.org/wiki/Point_in_T1_Space_is_Intersection_of_Open_Sets | [
"T1 Spaces",
"Set Intersection"
] | [
"Definition:Topological Space",
"Definition:T1 Space",
"Definition:Element",
"Definition:Singleton",
"Definition:Set Intersection",
"Definition:Open Set/Topology"
] | [
"Definition:Arbitrary",
"Definition:Set Intersection",
"Definition:Open Set/Topology",
"Definition:Arbitrary",
"Definition:Arbitrary",
"Definition:Arbitrary",
"Definition:Arbitrary",
"Definition:T1 Space",
"Definition:T1 Space",
"Definition:Arbitrary",
"Definition:Set Intersection",
"Definitio... |
proofwiki-23542 | Hermitian Conjugate is Adjoint | Let $\mathbf A$ be an $n \times m$ matrix over $\mathbb C$ that represents a linear transformation
:$\LL : V \to W$
where
:$V$ is $m$-dimensional
:$W$ is $n$-dimensional
in some orthonormal bases for $V$ and $W$.
Then, the adjoint $\LL^*$ is given by $\mathbf A^\dagger$, the Hermitian conjugate of $\mathbf A$. | Let $\mathbf x \in \mathbb C^m$ and $\mathbf y \in \mathbb C^n$ be vectors representing elements of $V$ and $W$, respectively.
Then:
{{begin-eqn}}
{{eqn | l = \innerprod {\mathbf A \mathbf x} {\mathbf y}
| r = \mathbf y^\dagger \paren {\mathbf A \mathbf x}
| c = {{Defof|Complex Vector Inner Product}}
}}
{{e... | Let $\mathbf A$ be an $n \times m$ [[Definition:Matrix|matrix]] over $\mathbb C$ that represents a [[Definition:Linear Transformation|linear transformation]]
:$\LL : V \to W$
where
:$V$ is $m$-[[Definition:Dimension of Vector Space|dimensional]]
:$W$ is $n$-[[Definition:Dimension of Vector Space|dimensional]]
in some [... | Let $\mathbf x \in \mathbb C^m$ and $\mathbf y \in \mathbb C^n$ be [[Definition:Vector|vectors]] representing elements of $V$ and $W$, respectively.
Then:
{{begin-eqn}}
{{eqn | l = \innerprod {\mathbf A \mathbf x} {\mathbf y}
| r = \mathbf y^\dagger \paren {\mathbf A \mathbf x}
| c = {{Defof|Complex Vector... | Hermitian Conjugate is Adjoint | https://proofwiki.org/wiki/Hermitian_Conjugate_is_Adjoint | https://proofwiki.org/wiki/Hermitian_Conjugate_is_Adjoint | [
"Adjoint Linear Transformations",
"Hermitian Conjugates"
] | [
"Definition:Matrix",
"Definition:Linear Transformation",
"Definition:Dimension of Vector Space",
"Definition:Dimension of Vector Space",
"Definition:Orthonormal Basis",
"Definition:Adjoint Linear Transformation",
"Definition:Hermitian Conjugate"
] | [
"Definition:Vector",
"Matrix Multiplication is Associative",
"Hermitian Conjugate of Matrix Product",
"Hermitian Conjugate is Involution"
] |
proofwiki-23543 | Normal Matrix is Normal Operator | A normal matrix is a normal operator. | Let $\mathbf A$ be a normal matrix.
Thus:
{{begin-eqn}}
{{eqn | l = \mathbf A \mathbf A^\dagger
| r = \mathbf A^\dagger \mathbf A
| c = {{Defof|Normal Matrix}}
}}
{{eqn | ll= \leadsto
| l = \mathbf A \mathbf A^*
| r = \mathbf A^* \mathbf A
| c = Hermitian Conjugate is Adjoint
}}
{{end-eqn}... | A [[Definition:Normal Matrix|normal matrix]] is a [[Definition:Normal Operator|normal operator]]. | Let $\mathbf A$ be a [[Definition:Normal Matrix|normal matrix]].
Thus:
{{begin-eqn}}
{{eqn | l = \mathbf A \mathbf A^\dagger
| r = \mathbf A^\dagger \mathbf A
| c = {{Defof|Normal Matrix}}
}}
{{eqn | ll= \leadsto
| l = \mathbf A \mathbf A^*
| r = \mathbf A^* \mathbf A
| c = [[Hermitian Co... | Normal Matrix is Normal Operator | https://proofwiki.org/wiki/Normal_Matrix_is_Normal_Operator | https://proofwiki.org/wiki/Normal_Matrix_is_Normal_Operator | [
"Normal Matrices",
"Normal Operators"
] | [
"Definition:Normal Matrix",
"Definition:Normal Operator"
] | [
"Definition:Normal Matrix",
"Hermitian Conjugate is Adjoint",
"Category:Normal Matrices",
"Category:Normal Operators"
] |
proofwiki-23544 | Hermitian Matrix is Normal | All Hermitian matrices are normal. | Let $\mathbf T$ be a Hermitian matrix.
We have:
{{begin-eqn}}
{{eqn | l = \mathbf T \mathbf T^\dagger
| r = \mathbf T \mathbf T
| c = {{Defof|Hermitian Matrix}}
}}
{{eqn | r = \mathbf T^\dagger \mathbf T
| c = {{Defof|Hermitian Matrix}}
}}
{{end-eqn}}
Thus, $\mathbf T$ is normal.
{{qed}}
Category:Herm... | All [[Definition:Hermitian Matrix|Hermitian matrices]] are [[Definition:Normal Matrix|normal]]. | Let $\mathbf T$ be a [[Definition:Hermitian Matrix|Hermitian matrix]].
We have:
{{begin-eqn}}
{{eqn | l = \mathbf T \mathbf T^\dagger
| r = \mathbf T \mathbf T
| c = {{Defof|Hermitian Matrix}}
}}
{{eqn | r = \mathbf T^\dagger \mathbf T
| c = {{Defof|Hermitian Matrix}}
}}
{{end-eqn}}
Thus, $\mathbf T$... | Hermitian Matrix is Normal | https://proofwiki.org/wiki/Hermitian_Matrix_is_Normal | https://proofwiki.org/wiki/Hermitian_Matrix_is_Normal | [
"Hermitian Matrices",
"Normal Matrices"
] | [
"Definition:Hermitian Matrix",
"Definition:Normal Matrix"
] | [
"Definition:Hermitian Matrix",
"Definition:Normal Matrix",
"Category:Hermitian Matrices",
"Category:Normal Matrices"
] |
proofwiki-23545 | Anti-Hermitian Matrix is Normal | All anti-Hermitian matrices are normal. | Let $\mathbf T$ be an anti-Hermitian matrix.
Then:
{{begin-eqn}}
{{eqn | l = \mathbf T \mathbf T^\dagger
| r = \mathbf T \paren {-\mathbf T}
| c = {{Defof|Anti-Hermitian Matrix}}
}}
{{eqn | r = -\mathbf T \mathbf T
| c = Matrix Multiplication is Homogeneous of Degree 1
}}
{{eqn | r = \mathbf T^\dagger... | All [[Definition:Anti-Hermitian Matrix|anti-Hermitian matrices]] are [[Definition:Normal Matrix|normal]]. | Let $\mathbf T$ be an [[Definition:Anti-Hermitian Matrix|anti-Hermitian matrix]].
Then:
{{begin-eqn}}
{{eqn | l = \mathbf T \mathbf T^\dagger
| r = \mathbf T \paren {-\mathbf T}
| c = {{Defof|Anti-Hermitian Matrix}}
}}
{{eqn | r = -\mathbf T \mathbf T
| c = [[Matrix Multiplication is Homogeneous of D... | Anti-Hermitian Matrix is Normal | https://proofwiki.org/wiki/Anti-Hermitian_Matrix_is_Normal | https://proofwiki.org/wiki/Anti-Hermitian_Matrix_is_Normal | [
"Normal Matrices",
"Anti-Hermitian Matrices"
] | [
"Definition:Anti-Hermitian Matrix",
"Definition:Normal Matrix"
] | [
"Definition:Anti-Hermitian Matrix",
"Matrix Multiplication is Homogeneous of Degree 1",
"Definition:Normal Matrix",
"Category:Normal Matrices",
"Category:Anti-Hermitian Matrices"
] |
proofwiki-23546 | Unitary Matrix is Normal | All unitary matrices are normal. | Let $\mathbf U$ be a unitary matrix of order $n$.
Then:
{{begin-eqn}}
{{eqn | l = \mathbf U \mathbf U^\dagger
| r = \mathbf U \mathbf U^{-1}
| c = {{Defof|Unitary Matrix}}
}}
{{eqn | r = \mathbf I_n
| c = {{Defof|Inverse Matrix}}
}}
{{eqn | r = \mathbf U^{-1} \mathbf U
| c = {{Defof|Inverse Matr... | All [[Definition:Unitary Matrix|unitary matrices]] are [[Definition:Normal Matrix|normal]]. | Let $\mathbf U$ be a [[Definition:Unitary Matrix|unitary matrix]] of [[Definition:Order of Matrix|order]] $n$.
Then:
{{begin-eqn}}
{{eqn | l = \mathbf U \mathbf U^\dagger
| r = \mathbf U \mathbf U^{-1}
| c = {{Defof|Unitary Matrix}}
}}
{{eqn | r = \mathbf I_n
| c = {{Defof|Inverse Matrix}}
}}
{{eqn |... | Unitary Matrix is Normal | https://proofwiki.org/wiki/Unitary_Matrix_is_Normal | https://proofwiki.org/wiki/Unitary_Matrix_is_Normal | [
"Unitary Matrices",
"Normal Matrices"
] | [
"Definition:Unitary Matrix",
"Definition:Normal Matrix"
] | [
"Definition:Unitary Matrix",
"Definition:Matrix/Order",
"Definition:Normal Matrix",
"Category:Unitary Matrices",
"Category:Normal Matrices"
] |
proofwiki-23547 | Eigenvalues of Unitary Matrix have Unit Modulus | Let $\mathbf U$ be a unitary matrix.
Let $\lambda$ be an eigenvalue of $\mathbf U$.
Then:
:$\size \lambda = 1$ | Let $\mathbf v$ be an eigenvector of $\mathbf U$ associated with $\lambda$.
Then:
{{begin-eqn}}
{{eqn | l = \norm {\mathbf U \mathbf v}^2
| r = \innerprod {\mathbf U \mathbf v} {\mathbf U \mathbf v}
| c = {{Defof|Inner Product Norm}}
}}
{{eqn | r = \innerprod {\lambda \mathbf v} {\lambda \mathbf v}
| ... | Let $\mathbf U$ be a [[Definition:Unitary Matrix|unitary matrix]].
Let $\lambda$ be an [[Definition:Eigenvalue|eigenvalue]] of $\mathbf U$.
Then:
:$\size \lambda = 1$ | Let $\mathbf v$ be an [[Definition:Eigenvector|eigenvector]] of $\mathbf U$ associated with $\lambda$.
Then:
{{begin-eqn}}
{{eqn | l = \norm {\mathbf U \mathbf v}^2
| r = \innerprod {\mathbf U \mathbf v} {\mathbf U \mathbf v}
| c = {{Defof|Inner Product Norm}}
}}
{{eqn | r = \innerprod {\lambda \mathbf v} ... | Eigenvalues of Unitary Matrix have Unit Modulus | https://proofwiki.org/wiki/Eigenvalues_of_Unitary_Matrix_have_Unit_Modulus | https://proofwiki.org/wiki/Eigenvalues_of_Unitary_Matrix_have_Unit_Modulus | [
"Unitary Matrices",
"Eigenvalues of Square Matrices"
] | [
"Definition:Unitary Matrix",
"Definition:Eigenvalue"
] | [
"Definition:Eigenvector",
"Inner Product is Sesquilinear",
"Modulus in Terms of Conjugate",
"Hermitian Conjugate is Adjoint",
"Definition:Eigenvector",
"Definition:Inner Product",
"Category:Unitary Matrices",
"Category:Eigenvalues of Square Matrices"
] |
proofwiki-23548 | Weakly Sigma-Locally Compact Space is Weakly Locally Compact | Let $T = \struct {S, \tau}$ be a weakly $\sigma$-locally compact space.
Then $T$ is weakly locally compact. | {{Recall|Weakly Sigma-Locally Compact Space|weakly $\sigma$-locally compact space}}
{{:Definition:Weakly Sigma-Locally Compact Space}}
The result follows by definition.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Weakly Sigma-Locally Compact Space|weakly $\sigma$-locally compact space]].
Then $T$ is [[Definition:Weakly Locally Compact Space|weakly locally compact]]. | {{Recall|Weakly Sigma-Locally Compact Space|weakly $\sigma$-locally compact space}}
{{:Definition:Weakly Sigma-Locally Compact Space}}
The result follows by definition.
{{qed}} | Weakly Sigma-Locally Compact Space is Weakly Locally Compact | https://proofwiki.org/wiki/Weakly_Sigma-Locally_Compact_Space_is_Weakly_Locally_Compact | https://proofwiki.org/wiki/Weakly_Sigma-Locally_Compact_Space_is_Weakly_Locally_Compact | [
"Weakly Sigma-Locally Compact Spaces",
"Weakly Locally Compact Spaces"
] | [
"Definition:Weakly Sigma-Locally Compact Space",
"Definition:Weakly Locally Compact Space"
] | [] |
proofwiki-23549 | Subcover of Open Cover is Open | Let $T = \struct {S, \tau}$ be a topological space.
Let $\CC$ be an open cover of $T$.
Let $\CC'$ be a subcover of $S$.
Then $\CC'$ is also an open cover of $T$. | {{Recall|Open Cover|open cover}}
{{:Definition:Open Cover}}
{{Recall|Subcover|subcover}}
{{:Definition:Subcover}}
Let $U \in \CC'$ be arbitrary.
By definition of subcover, $U \in \CC$.
By definition of open cover, $U$ is open in $T$.
As $U$ is arbitrary, it follows that all elements of $\CC$ are open in $T$.
Hence, by ... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $\CC$ be an [[Definition:Open Cover|open cover]] of $T$.
Let $\CC'$ be a [[Definition:Subcover|subcover]] of $S$.
Then $\CC'$ is also an [[Definition:Open Cover|open cover]] of $T$. | {{Recall|Open Cover|open cover}}
{{:Definition:Open Cover}}
{{Recall|Subcover|subcover}}
{{:Definition:Subcover}}
Let $U \in \CC'$ be [[Definition:Arbitrary|arbitrary]].
By definition of [[Definition:Subcover|subcover]], $U \in \CC$.
By definition of [[Definition:Open Cover|open cover]], $U$ is [[Definition:Open Se... | Subcover of Open Cover is Open | https://proofwiki.org/wiki/Subcover_of_Open_Cover_is_Open | https://proofwiki.org/wiki/Subcover_of_Open_Cover_is_Open | [
"Open Covers",
"Subcovers"
] | [
"Definition:Topological Space",
"Definition:Open Cover",
"Definition:Subcover",
"Definition:Open Cover"
] | [
"Definition:Arbitrary",
"Definition:Subcover",
"Definition:Open Cover",
"Definition:Open Set/Topology",
"Definition:Arbitrary",
"Definition:Element",
"Definition:Open Set/Topology",
"Definition:Open Cover",
"Category:Open Covers",
"Category:Subcovers"
] |
proofwiki-23550 | Product of Matrix and Standard Basis Vector is Column of Matrix | Let $\mathbb F$ be a field.
Let $\mathbf M$ be an $n \times m$ matrix over $\mathbb F$.
Let $\mathbf m_p$ be the $p^{th}$ column of $\mathbf M$.
Let $\mathbf e_p$ be the $p^{th}$ standard basis vector of $\mathbb F^m$ represented as a column vector.
Then:
:$\mathbf M \mathbf e_p = \mathbf m_p$ | By definition, since the order of $\mathbf M$ is $n \times m$ and the order of $\mathbf e_p$ is $m \times 1$, the order of $\mathbf M \mathbf e_p$ is $n \times 1$.
Thus, $\mathbf M \mathbf e_p$ is a column vector.
Let $\sqbrk {\mathbf M \mathbf e_p}_i$ be the $i^{th}$ component of $\mathbf M \mathbf e_p$.
Let $M_{ij}$ ... | Let $\mathbb F$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $\mathbf M$ be an $n \times m$ [[Definition:Matrix|matrix]] over $\mathbb F$.
Let $\mathbf m_p$ be the $p^{th}$ [[Definition:Column of Matrix|column]] of $\mathbf M$.
Let $\mathbf e_p$ be the $p^{th}$ [[Definition:Standard Ordered Basis on Vecto... | By [[Definition:Matrix Product (Conventional)|definition]], since the [[Definition:Order of Matrix|order]] of $\mathbf M$ is $n \times m$ and the [[Definition:Order of Matrix|order]] of $\mathbf e_p$ is $m \times 1$, the [[Definition:Order of Matrix|order]] of $\mathbf M \mathbf e_p$ is $n \times 1$.
Thus, $\mathbf M ... | Product of Matrix and Standard Basis Vector is Column of Matrix | https://proofwiki.org/wiki/Product_of_Matrix_and_Standard_Basis_Vector_is_Column_of_Matrix | https://proofwiki.org/wiki/Product_of_Matrix_and_Standard_Basis_Vector_is_Column_of_Matrix | [
"Conventional Matrix Multiplication",
"Standard Ordered Bases"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Matrix",
"Definition:Matrix/Column",
"Definition:Standard Ordered Basis/Vector Space",
"Definition:Column Matrix"
] | [
"Definition:Matrix Product (Conventional)",
"Definition:Matrix/Order",
"Definition:Matrix/Order",
"Definition:Matrix/Order",
"Definition:Column Matrix",
"Definition:Vector Quantity/Component",
"Definition:Matrix/Element",
"Definition:Vector Quantity/Component",
"Definition:Vector Quantity/Component"... |
proofwiki-23551 | Real Orthogonal Matrix is Unitary | All real orthogonal matrices are unitary. | Let $\mathbf R$ be a real orthogonal matrix.
Then:
{{begin-eqn}}
{{eqn | l = \mathbf R^\dagger
| r = \mathbf R^\intercal
| c = Hermitian Conjugate of Real Matrix is Transpose
}}
{{eqn | r = \mathbf R^{-1}
| c = {{Defof|Orthogonal Matrix}}
}}
{{end-eqn}}
{{qed}}
Category:Orthogonal Matrices
Category:Un... | All [[Definition:Real Matrix|real]] [[Definition:Orthogonal Matrix|orthogonal matrices]] are [[Definition:Unitary Matrix|unitary]]. | Let $\mathbf R$ be a [[Definition:Real Matrix|real]] [[Definition:Orthogonal Matrix|orthogonal matrix]].
Then:
{{begin-eqn}}
{{eqn | l = \mathbf R^\dagger
| r = \mathbf R^\intercal
| c = [[Hermitian Conjugate of Real Matrix is Transpose]]
}}
{{eqn | r = \mathbf R^{-1}
| c = {{Defof|Orthogonal Matrix}... | Real Orthogonal Matrix is Unitary | https://proofwiki.org/wiki/Real_Orthogonal_Matrix_is_Unitary | https://proofwiki.org/wiki/Real_Orthogonal_Matrix_is_Unitary | [
"Orthogonal Matrices",
"Unitary Matrices"
] | [
"Definition:Real Matrix",
"Definition:Orthogonal Matrix",
"Definition:Unitary Matrix"
] | [
"Definition:Real Matrix",
"Definition:Orthogonal Matrix",
"Hermitian Conjugate of Real Matrix is Transpose",
"Category:Orthogonal Matrices",
"Category:Unitary Matrices"
] |
proofwiki-23552 | Real Orthogonal Matrix is Normal | All real orthogonal matrices are normal. | Follows directly from Real Orthogonal Matrix is Unitary and Unitary Matrix is Normal.
Category:Orthogonal Matrices
Category:Normal Matrices
jkm9uk6mt4vxvk5irw8c2dikrq10uo1 | All [[Definition:Real Matrix|real]] [[Definition:Orthogonal Matrix|orthogonal matrices]] are [[Definition:Normal Matrix|normal]]. | Follows directly from [[Real Orthogonal Matrix is Unitary]] and [[Unitary Matrix is Normal]].
[[Category:Orthogonal Matrices]]
[[Category:Normal Matrices]]
jkm9uk6mt4vxvk5irw8c2dikrq10uo1 | Real Orthogonal Matrix is Normal | https://proofwiki.org/wiki/Real_Orthogonal_Matrix_is_Normal | https://proofwiki.org/wiki/Real_Orthogonal_Matrix_is_Normal | [
"Orthogonal Matrices",
"Normal Matrices"
] | [
"Definition:Real Matrix",
"Definition:Orthogonal Matrix",
"Definition:Normal Matrix"
] | [
"Real Orthogonal Matrix is Unitary",
"Unitary Matrix is Normal",
"Category:Orthogonal Matrices",
"Category:Normal Matrices"
] |
proofwiki-23553 | Real Symmetric Matrix is Normal | All real symmetric matrices are normal. | Follows directly from Real Symmetric Matrix is Hermitian and Hermitian Matrix is Normal.
Category:Real Matrices
Category:Symmetric Matrices
Category:Normal Matrices
2bnl4xd6ljue8ogkehtv45glhcqvye1 | All [[Definition:Real Matrix|real]] [[Definition:Symmetric Matrix|symmetric matrices]] are [[Definition:Normal Matrix|normal]]. | Follows directly from [[Real Symmetric Matrix is Hermitian]] and [[Hermitian Matrix is Normal]].
[[Category:Real Matrices]]
[[Category:Symmetric Matrices]]
[[Category:Normal Matrices]]
2bnl4xd6ljue8ogkehtv45glhcqvye1 | Real Symmetric Matrix is Normal | https://proofwiki.org/wiki/Real_Symmetric_Matrix_is_Normal | https://proofwiki.org/wiki/Real_Symmetric_Matrix_is_Normal | [
"Real Matrices",
"Symmetric Matrices",
"Normal Matrices",
"Real Matrices",
"Symmetric Matrices",
"Normal Matrices"
] | [
"Definition:Real Matrix",
"Definition:Symmetric Matrix",
"Definition:Normal Matrix"
] | [
"Real Symmetric Matrix is Hermitian",
"Hermitian Matrix is Normal",
"Category:Real Matrices",
"Category:Symmetric Matrices",
"Category:Normal Matrices"
] |
proofwiki-23554 | Matrix Normality Preserved by Unitary Transformation | Let $\mathbf A$ and $\mathbf B$ be square matrices over $\mathbb C$.
Let $\mathbf A$ and $\mathbf B$ be similar via a unitary transformation, that is:
:$\mathbf B = \mathbf U \mathbf A \mathbf U^\dagger$
where
:$\mathbf U$ is a unitary matrix
:$\mathbf U^\dagger$ is the Hermitian conjugate of $\mathbf U$
Then, $\mathbf... | Let $\mathbf A$ be normal.
Then:
{{begin-eqn}}
{{eqn | l = \mathbf B \mathbf B^\dagger
| r = \paren {\mathbf U \mathbf A \mathbf U^\dagger} \paren {\mathbf U \mathbf A \mathbf U^\dagger}^\dagger
}}
{{eqn | r = \paren {\mathbf U \mathbf A \mathbf U^\dagger} \paren {\mathbf U \mathbf A^\dagger \mathbf U^\dagger}
... | Let $\mathbf A$ and $\mathbf B$ be [[Definition:Square Matrix|square matrices]] over $\mathbb C$.
Let $\mathbf A$ and $\mathbf B$ be [[Definition:Similar Matrices|similar]] via a [[Definition:Unitary Transformation|unitary transformation]], that is:
:$\mathbf B = \mathbf U \mathbf A \mathbf U^\dagger$
where
:$\mathbf ... | Let $\mathbf A$ be [[Definition:Normal Matrix|normal]].
Then:
{{begin-eqn}}
{{eqn | l = \mathbf B \mathbf B^\dagger
| r = \paren {\mathbf U \mathbf A \mathbf U^\dagger} \paren {\mathbf U \mathbf A \mathbf U^\dagger}^\dagger
}}
{{eqn | r = \paren {\mathbf U \mathbf A \mathbf U^\dagger} \paren {\mathbf U \mathbf A... | Matrix Normality Preserved by Unitary Transformation | https://proofwiki.org/wiki/Matrix_Normality_Preserved_by_Unitary_Transformation | https://proofwiki.org/wiki/Matrix_Normality_Preserved_by_Unitary_Transformation | [
"Matrix Similarity",
"Normal Matrices",
"Unitary Transformations"
] | [
"Definition:Matrix/Square Matrix",
"Definition:Matrix Similarity",
"Definition:Unitary Transformation",
"Definition:Unitary Matrix",
"Definition:Hermitian Conjugate",
"Definition:Normal Matrix",
"Definition:Normal Matrix"
] | [
"Definition:Normal Matrix",
"Hermitian Conjugate of Matrix Product",
"Hermitian Conjugate is Involution",
"Matrix Multiplication is Associative",
"Definition:Unitary Matrix",
"Definition:Normal Matrix",
"Definition:Unitary Matrix",
"Matrix Multiplication is Associative",
"Hermitian Conjugate of Matr... |
proofwiki-23555 | Finite Cover is Locally Finite | Let $S$ be a set.
Let $\CC$ be a finite cover for a set $S$.
Then $\CC$ is also a locally finite cover for $S$. | {{Recall|Locally Finite Cover|locally finite cover}}
{{:Definition:Locally Finite Cover}}
Let $\CC$ be a finite cover for a set $S$.
{{Recall|Finite Cover|finite cover}}
{{:Definition:Finite Cover}}
Let $x \in S$ be arbitrary.
From Topological Space is Neighborhood of all its Points, $S$ is a neighborhood of $x$.
As $\... | Let $S$ be a [[Definition:Set|set]].
Let $\CC$ be a [[Definition:Finite Cover|finite cover]] for a [[Definition:Set|set]] $S$.
Then $\CC$ is also a [[Definition:Locally Finite Cover|locally finite cover]] for $S$. | {{Recall|Locally Finite Cover|locally finite cover}}
{{:Definition:Locally Finite Cover}}
Let $\CC$ be a [[Definition:Finite Cover|finite cover]] for a [[Definition:Set|set]] $S$.
{{Recall|Finite Cover|finite cover}}
{{:Definition:Finite Cover}}
Let $x \in S$ be [[Definition:Arbitrary|arbitrary]].
From [[Topologica... | Finite Cover is Locally Finite | https://proofwiki.org/wiki/Finite_Cover_is_Locally_Finite | https://proofwiki.org/wiki/Finite_Cover_is_Locally_Finite | [
"Finite Covers",
"Locally Finite Covers"
] | [
"Definition:Set",
"Definition:Cover of Set/Finite",
"Definition:Set",
"Definition:Locally Finite Cover"
] | [
"Definition:Cover of Set/Finite",
"Definition:Set",
"Definition:Arbitrary",
"Topological Space is Neighborhood of all its Points",
"Definition:Neighborhood (Topology)/Point",
"Definition:Cover of Set/Finite",
"Definition:Finite Set",
"Definition:Element",
"Definition:Set Intersection",
"Definition... |
proofwiki-23556 | Characterization of Diagonalizable Matrices | Let $\mathbf A$ be a diagonalizable matrix of order $n$ over a field $\mathbb F$:
:$\mathbf A = \mathbf X \mathbf D \mathbf X^{-1}$
where
{{begin-itemize}}
{{item||$\mathbf X$ is a nonsingular matrix of order $n$}}
{{item||$\mathbf D$ is a diagonal matrix of order $n$.}}
{{end-itemize}}
Then:
{{begin-itemize}}
{{item|(... | We have that $\mathbf D$ is diagonal.
Let $d_i = \sqbrk {\mathbf D}_{ii}$ be the $i^{th}$ diagonal element of $\mathbf D$.
Let $\mathbf x_i$ be a column of $\mathbf X$.
Let $\mathbf e_i$ be the $i^{th}$ standard basis vector of $\mathbb F^n$.
Therefore:
{{begin-eqn}}
{{eqn | l = \mathbf A
| r = \mathbf X \mathbf ... | Let $\mathbf A$ be a [[Definition:Diagonalizable Matrix|diagonalizable matrix]] of [[Definition:Order of Matrix|order]] $n$ over a [[Definition:Field (Abstract Algebra)|field]] $\mathbb F$:
:$\mathbf A = \mathbf X \mathbf D \mathbf X^{-1}$
where
{{begin-itemize}}
{{item||$\mathbf X$ is a [[Definition:Nonsingular Matrix... | We have that $\mathbf D$ is [[Definition:Diagonal Matrix|diagonal]].
Let $d_i = \sqbrk {\mathbf D}_{ii}$ be the $i^{th}$ [[Definition:Main Diagonal|diagonal]] [[Definition:Element of Matrix|element]] of $\mathbf D$.
Let $\mathbf x_i$ be a [[Definition:Column of Matrix|column]] of $\mathbf X$.
Let $\mathbf e_i$ be th... | Characterization of Diagonalizable Matrices | https://proofwiki.org/wiki/Characterization_of_Diagonalizable_Matrices | https://proofwiki.org/wiki/Characterization_of_Diagonalizable_Matrices | [
"Diagonalizable Matrices",
"Eigenvalues of Square Matrices",
"Eigenvectors of Square Matrices"
] | [
"Definition:Diagonalizable Matrix",
"Definition:Matrix/Order",
"Definition:Field (Abstract Algebra)",
"Definition:Nonsingular Matrix",
"Definition:Matrix/Order",
"Definition:Diagonal Matrix",
"Definition:Matrix/Order",
"Definition:Matrix/Diagonal/Main",
"Definition:Matrix/Element",
"Definition:Eig... | [
"Definition:Diagonal Matrix",
"Definition:Matrix/Diagonal/Main",
"Definition:Matrix/Element",
"Definition:Matrix/Column",
"Definition:Standard Ordered Basis/Vector Space",
"Definition:Matrix Product (Conventional)/Post-Multiplication",
"Definition:Matrix/Column",
"Matrix Multiplication is Homogeneous ... |
proofwiki-23557 | Upper Triangular Normal Matrix is Diagonal | A square matrix that is both upper triangular and normal is diagonal. | Proof by induction.
For all $n \in \mathbb N_{>0}$, let $\map P n$ be the proposition:
:An $n \times n$ upper triangular normal matrix is diagonal | A [[Definition:Square Matrix|square matrix]] that is both [[Definition:Upper Triangular Matrix|upper triangular]] and [[Definition:Normal Matrix|normal]] is [[Definition:Diagonal Matrix|diagonal]]. | Proof by [[Definition:Principle of Mathematical Induction|induction]].
For all $n \in \mathbb N_{>0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:An $n \times n$ [[Definition:Upper Triangular Matrix|upper triangular]] [[Definition:Normal Matrix|normal]] [[Definition:Matrix|matrix]] is [[Definition:... | Upper Triangular Normal Matrix is Diagonal | https://proofwiki.org/wiki/Upper_Triangular_Normal_Matrix_is_Diagonal | https://proofwiki.org/wiki/Upper_Triangular_Normal_Matrix_is_Diagonal | [
"Triangular Normal Matrix is Diagonal",
"Normal Matrices",
"Upper Triangular Matrices",
"Diagonal Matrices",
"Proofs by Induction"
] | [
"Definition:Matrix/Square Matrix",
"Definition:Triangular Matrix/Upper Triangular Matrix",
"Definition:Normal Matrix",
"Definition:Diagonal Matrix"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Definition:Triangular Matrix/Upper Triangular Matrix",
"Definition:Normal Matrix",
"Definition:Matrix",
"Definition:Diagonal Matrix",
"Definition:Triangular Matrix/Upper Triangular Matrix",
"Definition:Diagonal Matrix",
"Definition:No... |
proofwiki-23558 | Spectral Theorem for Normal Matrices | Let $\mathbf A$ be a square matrix over $\mathbb C$.
Then $\mathbf A$ is normal {{Iff}} it is diagonalizable by a unitary transformation.
That is, we can write:
:$\mathbf A = \mathbf U \mathbf D \mathbf U^\dagger$
where
:$\mathbf U$ is a unitary matrix
:$\mathbf D$ is a diagonal matrix | === Necessary Case ===
Assume $\mathbf A$ is diagonalizable by a unitary transformation.
Then:
{{begin-eqn}}
{{eqn | l = \mathbf A \mathbf A^\dagger
| r = \paren {\mathbf U \mathbf D \mathbf U^\dagger} \paren {\mathbf U \mathbf D \mathbf U^\dagger}^\dagger
}}
{{eqn | r = \paren {\mathbf U \mathbf D \mathbf U^\dag... | Let $\mathbf A$ be a [[Definition:Square Matrix|square matrix]] over $\mathbb C$.
Then $\mathbf A$ is [[Definition:Normal Matrix|normal]] {{Iff}} it is [[Definition:Diagonalizable Matrix|diagonalizable]] by a [[Definition:Unitary Transformation|unitary transformation]].
That is, we can write:
:$\mathbf A = \mathbf U ... | === Necessary Case ===
Assume $\mathbf A$ is [[Definition:Diagonalizable Matrix|diagonalizable]] by a [[Definition:Unitary Transformation|unitary transformation]].
Then:
{{begin-eqn}}
{{eqn | l = \mathbf A \mathbf A^\dagger
| r = \paren {\mathbf U \mathbf D \mathbf U^\dagger} \paren {\mathbf U \mathbf D \mathbf... | Spectral Theorem for Normal Matrices | https://proofwiki.org/wiki/Spectral_Theorem_for_Normal_Matrices | https://proofwiki.org/wiki/Spectral_Theorem_for_Normal_Matrices | [
"Spectral Theorems",
"Normal Matrices",
"Diagonalizable Matrices",
"Unitary Transformations"
] | [
"Definition:Matrix/Square Matrix",
"Definition:Normal Matrix",
"Definition:Diagonalizable Matrix",
"Definition:Unitary Transformation",
"Definition:Unitary Matrix",
"Definition:Diagonal Matrix"
] | [
"Definition:Diagonalizable Matrix",
"Definition:Unitary Transformation",
"Hermitian Conjugate of Matrix Product",
"Hermitian Conjugate is Involution",
"Matrix Multiplication is Associative",
"Matrix Multiplication on Diagonal Matrices is Commutative",
"Matrix Multiplication is Associative",
"Hermitian... |
proofwiki-23559 | Spectral Theorem for Unitary Matrices | Let $\mathbf A$ be a square matrix.
Then, $\mathbf A$ is a unitary matrix {{Iff}} it is diagonalizable via a unitary transformation to a diagonal matrix whose diagonal elements are of unit modulus.
That is, we can write:
:$\mathbf A = \mathbf U \mathbf D \mathbf U^\dagger$
where
:$\mathbf U$ is a unitary matrix
:$\math... | === Necessary Case ===
Let $A$ be diagonalizable via a unitary transformation to a diagonal matrix whose diagonal elements are of unit modulus.
Let $d_i = \sqbrk {\mathbf D}_{ii}$ be the $i^{th}$ diagonal element of $\mathbf D$.
Then:
{{begin-eqn}}
{{eqn | l = \mathbf A^{-1}
| r = \paren {\mathbf U \mathbf D \mat... | Let $\mathbf A$ be a [[Definition:Square Matrix|square matrix]].
Then, $\mathbf A$ is a [[Definition:Unitary Matrix|unitary matrix]] {{Iff}} it is [[Definition:Diagonalizable Matrix|diagonalizable]] via a [[Definition:Unitary Transformation|unitary transformation]] to a [[Definition:Diagonal Matrix|diagonal matrix]] w... | === Necessary Case ===
Let $A$ be [[Definition:Diagonalizable Matrix|diagonalizable]] via a [[Definition:Unitary Transformation|unitary transformation]] to a [[Definition:Diagonal Matrix|diagonal matrix]] whose [[Definition:Main Diagonal|diagonal elements]] are of [[Definition:Unit (One)|unit]] [[Definition:Complex Mo... | Spectral Theorem for Unitary Matrices | https://proofwiki.org/wiki/Spectral_Theorem_for_Unitary_Matrices | https://proofwiki.org/wiki/Spectral_Theorem_for_Unitary_Matrices | [
"Spectral Theorems",
"Unitary Transformations",
"Diagonalizable Matrices"
] | [
"Definition:Matrix/Square Matrix",
"Definition:Unitary Matrix",
"Definition:Diagonalizable Matrix",
"Definition:Unitary Transformation",
"Definition:Diagonal Matrix",
"Definition:Matrix/Diagonal/Main",
"Definition:Unit (One)",
"Definition:Complex Modulus",
"Definition:Unitary Matrix",
"Definition:... | [
"Definition:Diagonalizable Matrix",
"Definition:Unitary Transformation",
"Definition:Diagonal Matrix",
"Definition:Matrix/Diagonal/Main",
"Definition:Unit (One)",
"Definition:Complex Modulus",
"Definition:Matrix/Diagonal/Main",
"Definition:Matrix/Element",
"Inverse of Matrix Product",
"Inverse of ... |
proofwiki-23560 | Eigenvalues of Anti-Hermitian Matrix are Imaginary | The eigenvalues of an anti-Hermitian matrix are imaginary. | Let $\mathbf H$ be an anti-Hermitian matrix.
Let $\lambda$ be an eigenvalue of $\mathbf H$.
Let $\mathbf v$ be a eigenvector of $\mathbf H$ associated with $\lambda$.
Then:
{{begin-eqn}}
{{eqn | l = \lambda \innerprod {\mathbf v} {\mathbf v}
| r = \innerprod {\lambda \mathbf v} {\mathbf v}
| c = Inner Produ... | The [[Definition:Eigenvalue|eigenvalues]] of an [[Definition:Anti-Hermitian Matrix|anti-Hermitian matrix]] are [[Definition:Wholly Imaginary|imaginary]]. | Let $\mathbf H$ be an [[Definition:Anti-Hermitian Matrix|anti-Hermitian matrix]].
Let $\lambda$ be an [[Definition:Eigenvalue|eigenvalue]] of $\mathbf H$.
Let $\mathbf v$ be a [[Definition:Eigenvector|eigenvector]] of $\mathbf H$ associated with $\lambda$.
Then:
{{begin-eqn}}
{{eqn | l = \lambda \innerprod {\mathbf ... | Eigenvalues of Anti-Hermitian Matrix are Imaginary | https://proofwiki.org/wiki/Eigenvalues_of_Anti-Hermitian_Matrix_are_Imaginary | https://proofwiki.org/wiki/Eigenvalues_of_Anti-Hermitian_Matrix_are_Imaginary | [
"Anti-Hermitian Matrices",
"Eigenvalues of Square Matrices"
] | [
"Definition:Eigenvalue",
"Definition:Anti-Hermitian Matrix",
"Definition:Complex Number/Wholly Imaginary"
] | [
"Definition:Anti-Hermitian Matrix",
"Definition:Eigenvalue",
"Definition:Eigenvector",
"Inner Product is Sesquilinear",
"Hermitian Conjugate is Adjoint",
"Inner Product is Sesquilinear",
"Definition:Eigenvector",
"Definition:Zero Vector",
"Definition:Inner Product",
"Complex Number equals Negative... |
proofwiki-23561 | Spectral Theorem for Anti-Hermitian Matrices | Let $\mathbf A$ be a square matrix over $\mathbb C$.
Then $\mathbf A$ is anti-Hermitian {{Iff}} it is diagonalizable to an imaginary diagonal matrix via a unitary transformation.
That is, we can write:
:$\mathbf A = \mathbf U \mathbf D \mathbf U^\dagger$
where:
:$\mathbf D$ is an imaginary diagonal matrix
:$\mathbf U$ ... | === Necessary Case ===
Let $\mathbf A$ be diagonalizable to a imaginary diagonal matrix via a unitary transformation.
Then:
{{begin-eqn}}
{{eqn | l = \mathbf A^\dagger
| r = \paren {\mathbf U \mathbf D \mathbf U^\dagger}^\dagger
}}
{{eqn | r = \mathbf U \mathbf D^\dagger \mathbf U^\dagger
| c = Hermitian Co... | Let $\mathbf A$ be a [[Definition:Square Matrix|square matrix]] over $\mathbb C$.
Then $\mathbf A$ is [[Definition:Anti-Hermitian Matrix|anti-Hermitian]] {{Iff}} it is [[Definition:Diagonalizable Matrix|diagonalizable]] to an [[Definition:Wholly Imaginary|imaginary]] [[Definition:Diagonal Matrix|diagonal matrix]] via ... | === Necessary Case ===
Let $\mathbf A$ be [[Definition:Diagonalizable Matrix|diagonalizable]] to a [[Definition:Wholly Imaginary|imaginary]] [[Definition:Diagonal Matrix|diagonal matrix]] via a [[Definition:Unitary Transformation|unitary transformation]].
Then:
{{begin-eqn}}
{{eqn | l = \mathbf A^\dagger
| r = ... | Spectral Theorem for Anti-Hermitian Matrices | https://proofwiki.org/wiki/Spectral_Theorem_for_Anti-Hermitian_Matrices | https://proofwiki.org/wiki/Spectral_Theorem_for_Anti-Hermitian_Matrices | [
"Spectral Theorems",
"Anti-Hermitian Matrices",
"Diagonalizable Matrices",
"Unitary Transformations"
] | [
"Definition:Matrix/Square Matrix",
"Definition:Anti-Hermitian Matrix",
"Definition:Diagonalizable Matrix",
"Definition:Complex Number/Wholly Imaginary",
"Definition:Diagonal Matrix",
"Definition:Unitary Transformation",
"Definition:Complex Number/Wholly Imaginary",
"Definition:Diagonal Matrix",
"Def... | [
"Definition:Diagonalizable Matrix",
"Definition:Complex Number/Wholly Imaginary",
"Definition:Diagonal Matrix",
"Definition:Unitary Transformation",
"Hermitian Conjugate of Matrix Product",
"Hermitian Conjugate is Involution",
"Definition:Complex Number/Wholly Imaginary",
"Definition:Diagonal Matrix",... |
proofwiki-23562 | Real Anti-Symmetric Matrix is Anti-Hermitian | A real anti-symmetric matrix is anti-Hermitian. | Let $\mathbf A$ be a real anti-symmetric matrix.
Then:
{{begin-eqn}}
{{eqn | l = \mathbf A^\dagger
| r = \mathbf A^\intercal
| c = $\mathbf A$ is real
}}
{{eqn | r = -\mathbf A
| c = {{Defof|Anti-Symmetric Matrix}}
}}
{{end-eqn}}
Therefore, $\mathbf A$ is anti-Hermitian.
{{qed}}
Category:Antisymmetric... | A [[Definition:Real Matrix|real]] [[Definition:Anti-Symmetric Matrix|anti-symmetric matrix]] is [[Definition:Anti-Hermitian Matrix|anti-Hermitian]]. | Let $\mathbf A$ be a [[Definition:Real Matrix|real]] [[Definition:Anti-Symmetric Matrix|anti-symmetric matrix]].
Then:
{{begin-eqn}}
{{eqn | l = \mathbf A^\dagger
| r = \mathbf A^\intercal
| c = $\mathbf A$ is [[Definition:Real Matrix|real]]
}}
{{eqn | r = -\mathbf A
| c = {{Defof|Anti-Symmetric Matr... | Real Anti-Symmetric Matrix is Anti-Hermitian | https://proofwiki.org/wiki/Real_Anti-Symmetric_Matrix_is_Anti-Hermitian | https://proofwiki.org/wiki/Real_Anti-Symmetric_Matrix_is_Anti-Hermitian | [
"Antisymmetric Matrices",
"Anti-Hermitian Matrices",
"Real Matrices"
] | [
"Definition:Real Matrix",
"Definition:Antisymmetric Matrix",
"Definition:Anti-Hermitian Matrix"
] | [
"Definition:Real Matrix",
"Definition:Antisymmetric Matrix",
"Definition:Real Matrix",
"Definition:Anti-Hermitian Matrix",
"Category:Antisymmetric Matrices",
"Category:Anti-Hermitian Matrices",
"Category:Real Matrices"
] |
proofwiki-23563 | Eigenvalues of Real Anti-Symmetric Matrix are Imaginary | The eigenvalues of a real anti-symmetric matrix are imaginary. | Follows directly from Real Anti-Symmetric Matrix is Anti-Hermitian and Eigenvalues of Anti-Hermitian Matrix are Imaginary.
{{qed}}
Category:Real Matrices
Category:Antisymmetric Matrices
Category:Anti-Hermitian Matrices
Category:Eigenvalues of Square Matrices
Category:Eigenvalues of Real Square Matrices
d2d60pq8lcwy5wh6... | The [[Definition:Eigenvalue|eigenvalues]] of a [[Definition:Real Matrix|real]] [[Definition:Anti-Symmetric Matrix|anti-symmetric matrix]] are [[Definition:Wholly Imaginary|imaginary]]. | Follows directly from [[Real Anti-Symmetric Matrix is Anti-Hermitian]] and [[Eigenvalues of Anti-Hermitian Matrix are Imaginary]].
{{qed}}
[[Category:Real Matrices]]
[[Category:Antisymmetric Matrices]]
[[Category:Anti-Hermitian Matrices]]
[[Category:Eigenvalues of Square Matrices]]
[[Category:Eigenvalues of Real Squa... | Eigenvalues of Real Anti-Symmetric Matrix are Imaginary | https://proofwiki.org/wiki/Eigenvalues_of_Real_Anti-Symmetric_Matrix_are_Imaginary | https://proofwiki.org/wiki/Eigenvalues_of_Real_Anti-Symmetric_Matrix_are_Imaginary | [
"Real Matrices",
"Antisymmetric Matrices",
"Anti-Hermitian Matrices",
"Eigenvalues of Square Matrices",
"Eigenvalues of Real Square Matrices"
] | [
"Definition:Eigenvalue",
"Definition:Real Matrix",
"Definition:Antisymmetric Matrix",
"Definition:Complex Number/Wholly Imaginary"
] | [
"Real Anti-Symmetric Matrix is Anti-Hermitian",
"Eigenvalues of Anti-Hermitian Matrix are Imaginary",
"Category:Real Matrices",
"Category:Antisymmetric Matrices",
"Category:Anti-Hermitian Matrices",
"Category:Eigenvalues of Square Matrices",
"Category:Eigenvalues of Real Square Matrices"
] |
proofwiki-23564 | Spectral Theorem for Real Anti-Symmetric Matrices | Let $\mathbf A$ be a square matrix over $\R$.
Then $\mathbf A$ is anti-symmetric {{Iff}} it is diagonalizable to an imaginary diagonal matrix via a unitary transformation.
That is, we can write:
:$\mathbf A = \mathbf U \mathbf D \mathbf U^\dagger$
where:
:$\mathbf D$ is an imaginary diagonal matrix
:$\mathbf U$ is a un... | === Necessary Case ===
Let $\mathbf A$ be diagonalizable to an imaginary diagonal matrix via a unitary transformation.
From Spectral Theorem for Anti-Hermitian Matrices, $\mathbf A$ is anti-Hermitian.
Thus:
:$\mathbf A^\dagger = -\mathbf A$
Since $\mathbf A$ is real, $\mathbf A^\dagger = \mathbf A^\intercal$.
Thus:
:$\... | Let $\mathbf A$ be a [[Definition:Square Matrix|square matrix]] over $\R$.
Then $\mathbf A$ is [[Definition:Anti-Symmetric Matrix|anti-symmetric]] {{Iff}} it is [[Definition:Diagonalizable Matrix|diagonalizable]] to an [[Definition:Wholly Imaginary|imaginary]] [[Definition:Diagonal Matrix|diagonal matrix]] via a [[Def... | === Necessary Case ===
Let $\mathbf A$ be [[Definition:Diagonalizable Matrix|diagonalizable]] to an [[Definition:Wholly Imaginary|imaginary]] [[Definition:Diagonal Matrix|diagonal matrix]] via a [[Definition:Unitary Transformation|unitary transformation]].
From [[Spectral Theorem for Anti-Hermitian Matrices]], $\math... | Spectral Theorem for Real Anti-Symmetric Matrices | https://proofwiki.org/wiki/Spectral_Theorem_for_Real_Anti-Symmetric_Matrices | https://proofwiki.org/wiki/Spectral_Theorem_for_Real_Anti-Symmetric_Matrices | [
"Spectral Theorems",
"Antisymmetric Matrices",
"Diagonalizable Matrices",
"Unitary Transformations"
] | [
"Definition:Matrix/Square Matrix",
"Definition:Antisymmetric Matrix",
"Definition:Diagonalizable Matrix",
"Definition:Complex Number/Wholly Imaginary",
"Definition:Diagonal Matrix",
"Definition:Unitary Transformation",
"Definition:Complex Number/Wholly Imaginary",
"Definition:Diagonal Matrix",
"Defi... | [
"Definition:Diagonalizable Matrix",
"Definition:Complex Number/Wholly Imaginary",
"Definition:Diagonal Matrix",
"Definition:Unitary Transformation",
"Spectral Theorem for Anti-Hermitian Matrices",
"Definition:Anti-Hermitian Matrix",
"Definition:Real Matrix",
"Definition:Antisymmetric Matrix",
"Defin... |
proofwiki-23565 | Eigenvalues of Real Orthogonal Matrix have Unit Modulus and occur in Conjugate Pairs | Let $\mathbf R$ be a real orthogonal matrix.
Let $\lambda$ be an eigenvalue of $\mathbf R$.
Then:
:$\size \lambda = 1$
and
:$\overline \lambda$ is also an eigenvalue | From Real Orthogonal Matrix is Unitary, $\mathbf R$ is unitary.
From Eigenvalues of Unitary Matrix have Unit Modulus:
:$\size \lambda = 1$
We have that $\mathbf R$ is real.
From Complex Eigenvalues of Real Matrix occur in Conjugate Pairs, $\overline {\mathbf \lambda}$ is also an eigenvalue.
{{qed}}
Category:Orthogonal ... | Let $\mathbf R$ be a [[Definition:Real Matrix|real]] [[Definition:Orthogonal Matrix|orthogonal matrix]].
Let $\lambda$ be an [[Definition:Eigenvalue|eigenvalue]] of $\mathbf R$.
Then:
:$\size \lambda = 1$
and
:$\overline \lambda$ is also an [[Definition:Eigenvalue|eigenvalue]] | From [[Real Orthogonal Matrix is Unitary]], $\mathbf R$ is [[Definition:Unitary Matrix|unitary]].
From [[Eigenvalues of Unitary Matrix have Unit Modulus]]:
:$\size \lambda = 1$
We have that $\mathbf R$ is [[Definition:Real Matrix|real]].
From [[Complex Eigenvalues of Real Matrix occur in Conjugate Pairs]], $\overli... | Eigenvalues of Real Orthogonal Matrix have Unit Modulus and occur in Conjugate Pairs | https://proofwiki.org/wiki/Eigenvalues_of_Real_Orthogonal_Matrix_have_Unit_Modulus_and_occur_in_Conjugate_Pairs | https://proofwiki.org/wiki/Eigenvalues_of_Real_Orthogonal_Matrix_have_Unit_Modulus_and_occur_in_Conjugate_Pairs | [
"Orthogonal Matrices",
"Eigenvalues of Square Matrices",
"Eigenvalues of Real Square Matrices"
] | [
"Definition:Real Matrix",
"Definition:Orthogonal Matrix",
"Definition:Eigenvalue",
"Definition:Eigenvalue"
] | [
"Real Orthogonal Matrix is Unitary",
"Definition:Unitary Matrix",
"Eigenvalues of Unitary Matrix have Unit Modulus",
"Definition:Real Matrix",
"Complex Eigenvalues of Real Matrix occur in Conjugate Pairs",
"Definition:Eigenvalue",
"Category:Orthogonal Matrices",
"Category:Eigenvalues of Square Matrice... |
proofwiki-23566 | Spectral Theorem for Real Orthogonal Matrices | Let $\mathbf A$ be a square matrix over $\R$.
Then, $\mathbf A$ is orthogonal {{Iff}} it is diagonalizable via a unitary transformation to a diagonal matrix whose diagonal elements are of unit modulus which occur in conjugate pairs.
That is, we can write:
:$\mathbf A = \mathbf U \mathbf D \mathbf U^\dagger$
where
:$\ma... | === Necessary Case ===
Let $\mathbf A$ be a real matrix that is diagonalizable via a unitary transformation to a diagonal matrix whose diagonal elements are of unit modulus which occur in conjugate pairs.
From Spectral Theorem for Unitary Matrices, $\mathbf A$ is unitary.
Since $\mathbf A$ is real:
{{begin-eqn}}
{{eqn ... | Let $\mathbf A$ be a [[Definition:Square Matrix|square matrix]] over $\R$.
Then, $\mathbf A$ is [[Definition:Orthogonal Matrix|orthogonal]] {{Iff}} it is [[Definition:Diagonalizable Matrix|diagonalizable]] via a [[Definition:Unitary Transformation|unitary transformation]] to a [[Definition:Diagonal Matrix|diagonal mat... | === Necessary Case ===
Let $\mathbf A$ be a [[Definition:Real Matrix|real matrix]] that is [[Definition:Diagonalizable Matrix|diagonalizable]] via a [[Definition:Unitary Transformation|unitary transformation]] to a [[Definition:Diagonal Matrix|diagonal matrix]] whose [[Definition:Main Diagonal|diagonal elements]] are ... | Spectral Theorem for Real Orthogonal Matrices | https://proofwiki.org/wiki/Spectral_Theorem_for_Real_Orthogonal_Matrices | https://proofwiki.org/wiki/Spectral_Theorem_for_Real_Orthogonal_Matrices | [
"Spectral Theorems",
"Orthogonal Matrices",
"Diagonalizable Matrices",
"Unitary Transformations"
] | [
"Definition:Matrix/Square Matrix",
"Definition:Orthogonal Matrix",
"Definition:Diagonalizable Matrix",
"Definition:Unitary Transformation",
"Definition:Diagonal Matrix",
"Definition:Matrix/Diagonal/Main",
"Definition:Unit (One)",
"Definition:Complex Modulus",
"Definition:Complex Conjugate",
"Defin... | [
"Definition:Real Matrix",
"Definition:Diagonalizable Matrix",
"Definition:Unitary Transformation",
"Definition:Diagonal Matrix",
"Definition:Matrix/Diagonal/Main",
"Definition:Unit (One)",
"Definition:Complex Modulus",
"Definition:Complex Conjugate",
"Spectral Theorem for Unitary Matrices",
"Defin... |
proofwiki-23567 | Complex Eigenvalues of Real Matrix occur in Conjugate Pairs | The complex eigenvalues of a real matrix occur in conjugate pairs.
That is, if $\lambda$ is a complex eigenvalues of a real matrix, then so is the complex conjugate $\overline \lambda$. | By definition, the eigenvalues of a matrix are the roots of its characteristic polynomial.
The characteristic polynomial of a real matrix has real coefficients.
From Complex Roots of Polynomial with Real Coefficients occur in Conjugate Pairs, the complex eigenvalues come in conjugate pairs.
{{qed}}
Category:Eigenvalues... | The [[Definition:Complex Number|complex]] [[Definition:Eigenvalue|eigenvalues]] of a [[Definition:Real Matrix|real matrix]] occur in [[Definition:Complex Conjugate|conjugate]] pairs.
That is, if $\lambda$ is a [[Definition:Complex Number|complex]] [[Definition:Eigenvalue|eigenvalues]] of a [[Definition:Real Matrix|rea... | By definition, the [[Definition:Eigenvalue|eigenvalues]] of a [[Definition:Matrix|matrix]] are the [[Definition:Root of Polynomial|roots]] of its [[Definition:Characteristic Polynomial of Matrix|characteristic polynomial]].
The [[Definition:Characteristic Polynomial of Matrix|characteristic polynomial]] of a [[Definit... | Complex Eigenvalues of Real Matrix occur in Conjugate Pairs | https://proofwiki.org/wiki/Complex_Eigenvalues_of_Real_Matrix_occur_in_Conjugate_Pairs | https://proofwiki.org/wiki/Complex_Eigenvalues_of_Real_Matrix_occur_in_Conjugate_Pairs | [
"Eigenvalues of Square Matrices",
"Real Matrices"
] | [
"Definition:Complex Number",
"Definition:Eigenvalue",
"Definition:Real Matrix",
"Definition:Complex Conjugate",
"Definition:Complex Number",
"Definition:Eigenvalue",
"Definition:Real Matrix",
"Definition:Complex Conjugate"
] | [
"Definition:Eigenvalue",
"Definition:Matrix",
"Definition:Root of Polynomial",
"Definition:Characteristic Polynomial of Matrix",
"Definition:Characteristic Polynomial of Matrix",
"Definition:Real Matrix",
"Definition:Real Number",
"Definition:Coefficient of Polynomial",
"Complex Roots of Polynomial ... |
proofwiki-23568 | Expected Number of Rolls until Two Consecutive Heads | Let $0 < p \le 1$.
Consider a coin such that we toss a head with probability $p$.
We repeatedly toss this coin until we observe two consecutive heads.
The expected number of tosses until we observe two consecutive heads is:
:$\ds \frac 1 p + \frac 1 {p^2}$ | Let $E$ be the expected number of tosses until we observe two consecutive heads.
Let $E_H$ be the expected number of further tosses until we observe two consecutive heads given that our previous toss is a head.
Let $q = 1 - p$.
By the Total Expectation Theorem, we have:
:$E = p \expect {\text {number of tosses until t... | Let $0 < p \le 1$.
Consider a [[Definition:Coin|coin]] such that we [[Definition:Coin-Tossing|toss]] a [[Definition:Head of Coin|head]] with [[Definition:Probability|probability]] $p$.
We repeatedly toss this [[Definition:Coin|coin]] until we observe two consecutive [[Definition:Head of Coin|heads]].
The [[Definiti... | Let $E$ be the [[Definition:Expected Value|expected number]] of [[Definition:Coin-Tossing|tosses]] until we observe two consecutive [[Definition:Head of Coin|heads]].
Let $E_H$ be the [[Definition:Expected Value|expected number]] of further [[Definition:Coin-Tossing|tosses]] until we observe two consecutive [[Definiti... | Expected Number of Rolls until Two Consecutive Heads | https://proofwiki.org/wiki/Expected_Number_of_Rolls_until_Two_Consecutive_Heads | https://proofwiki.org/wiki/Expected_Number_of_Rolls_until_Two_Consecutive_Heads | [
"Combinatorics",
"Coins"
] | [
"Definition:Coin",
"Definition:Coin/Coin-Tossing",
"Definition:Coin/Head",
"Definition:Probability",
"Definition:Coin",
"Definition:Coin/Head",
"Definition:Expectation",
"Definition:Coin/Coin-Tossing",
"Definition:Coin/Head"
] | [
"Definition:Expectation",
"Definition:Coin/Coin-Tossing",
"Definition:Coin/Head",
"Definition:Expectation",
"Definition:Coin/Coin-Tossing",
"Definition:Coin/Head",
"Total Expectation Theorem",
"Definition:Coin/Head",
"Definition:Expectation",
"Definition:Coin/Head",
"Definition:Coin/Tail",
"To... |
proofwiki-23569 | Probability that Uniformly Selected Subset contains Particular Element | Let $X$ be a finite set.
Let $\powerset X$ be the power set of $X$.
Give $\powerset X$ the discrete uniform distribution, so that:
:$\ds \map \Pr {\set A} = 2^{-n}$ for each $A \in \powerset X$.
Let $x \in X$.
Then:
:$\ds \map \Pr {\set {A \in \powerset X : x \in A} } = \frac 1 2$ | We count the cardinality of $\set {A \in \powerset X : x \in A}$.
All sets $C \in \set {A \in \powerset X : x \in A}$ can be written uniquely in the form $\set x \cup B$, where $B \in \powerset {A \setminus \set x}$.
In particular, $B = C \setminus \set x$.
From Cardinality of Power Set of Finite Set, we have $\card {... | Let $X$ be a [[Definition:Finite Set|finite set]].
Let $\powerset X$ be the [[Definition:Power Set|power set]] of $X$.
Give $\powerset X$ the [[Definition:Discrete Uniform Distribution|discrete uniform distribution]], so that:
:$\ds \map \Pr {\set A} = 2^{-n}$ for each $A \in \powerset X$.
Let $x \in X$.
Then:
:$\... | We count the [[Definition:Cardinality|cardinality]] of $\set {A \in \powerset X : x \in A}$.
All [[Definition:Set|sets]] $C \in \set {A \in \powerset X : x \in A}$ can be written uniquely in the form $\set x \cup B$, where $B \in \powerset {A \setminus \set x}$.
In particular, $B = C \setminus \set x$.
From [[Cardi... | Probability that Uniformly Selected Subset contains Particular Element | https://proofwiki.org/wiki/Probability_that_Uniformly_Selected_Subset_contains_Particular_Element | https://proofwiki.org/wiki/Probability_that_Uniformly_Selected_Subset_contains_Particular_Element | [
"Probability",
"Power Set"
] | [
"Definition:Finite Set",
"Definition:Power Set",
"Definition:Uniform Distribution/Discrete"
] | [
"Definition:Cardinality",
"Definition:Set",
"Cardinality of Power Set of Finite Set",
"Definition:Cardinality",
"Definition:Uniform Distribution/Discrete",
"Category:Probability",
"Category:Power Set"
] |
proofwiki-23570 | Expected Sum of Randomly Drawn Subset | Let $n$ be a positive integer.
Let $\sqbrk n = \set {1, 2, \ldots, n}$ be the set of integers between $1$ and $n$.
Equip $\powerset {\sqbrk n}$ with the discrete uniform distribution.
Let $\map \Sigma S$ be the sum of a subset $S \in \powerset {\sqbrk n}$.
Then:
:$\ds \expect {\map \Sigma S} = \frac {n \paren {n + 1} }... | For each $k \in \sqbrk n$, define $1_k : \powerset {\sqbrk n} \to \set {0, 1}$ by:
:$\ds \map {1_k} A = \begin{cases}1 & k \in A \\ 0 & k \not \in A\end{cases}$
for each $A \in \powerset {\sqbrk n}$.
We can then write:
:$\ds \map \Sigma S = \sum_{k \mathop = 1}^n k \map {1_k} S$
From Expectation is Linear, we have:
:$\... | Let $n$ be a [[Definition:Positive Integer|positive integer]].
Let $\sqbrk n = \set {1, 2, \ldots, n}$ be the [[Definition:Set|set]] of [[Definition:Integer|integers]] between $1$ and $n$.
Equip $\powerset {\sqbrk n}$ with the [[Definition:Discrete Uniform Distribution|discrete uniform distribution]].
Let $\map \Sig... | For each $k \in \sqbrk n$, define $1_k : \powerset {\sqbrk n} \to \set {0, 1}$ by:
:$\ds \map {1_k} A = \begin{cases}1 & k \in A \\ 0 & k \not \in A\end{cases}$
for each $A \in \powerset {\sqbrk n}$.
We can then write:
:$\ds \map \Sigma S = \sum_{k \mathop = 1}^n k \map {1_k} S$
From [[Expectation is Linear]], we hav... | Expected Sum of Randomly Drawn Subset | https://proofwiki.org/wiki/Expected_Sum_of_Randomly_Drawn_Subset | https://proofwiki.org/wiki/Expected_Sum_of_Randomly_Drawn_Subset | [
"Probability",
"Power Set"
] | [
"Definition:Positive/Integer",
"Definition:Set",
"Definition:Integer",
"Definition:Uniform Distribution/Discrete",
"Definition:Sum",
"Definition:Subset"
] | [
"Expectation is Linear",
"Integral of Characteristic Function",
"Probability that Uniformly Selected Subset contains Particular Element",
"Sum of Arithmetic Sequence",
"Category:Probability",
"Category:Power Set"
] |
proofwiki-23571 | Expected Number of Trials Until N Consecutive Successes | Let $0 < p \le 1$.
Let $N$ be a positive integer.
We conduct an experiment with probability of success $p$.
We run this experiment until we see $N$ consecutive successes.
Then the expected number of trials until we see $N$ consecutive successes is:
:$\ds \sum_{k \mathop = 1}^N p^{-k} = \frac 1 p + \frac 1 {p^2} + \ldo... | Let $E_N$ be the expected number of trials until we see $N$ consecutive successes.
We look to form a recurrence relation for $E_N$ with $N > 1$.
First, we run the experiment until we see $N$ consecutive successes.
In expectation, this will take $E_{N - 1}$ trials.
There is a probability $p$ that our next trial will su... | Let $0 < p \le 1$.
Let $N$ be a [[Definition:Positive Integer|positive integer]].
We conduct an [[Definition:Experiment|experiment]] with [[Definition:Probability|probability]] of success $p$.
We run this [[Definition:Experiment|experiment]] until we see $N$ consecutive successes.
Then the [[Definition:Expected V... | Let $E_N$ be the [[Definition:Expected Value|expected number]] of [[Definition:Trial|trials]] until we see $N$ consecutive successes.
We look to form a [[Definition:Recurrence Relation|recurrence relation]] for $E_N$ with $N > 1$.
First, we run the [[Definition:Experiment|experiment]] until we see $N$ consecutive su... | Expected Number of Trials Until N Consecutive Successes | https://proofwiki.org/wiki/Expected_Number_of_Trials_Until_N_Consecutive_Successes | https://proofwiki.org/wiki/Expected_Number_of_Trials_Until_N_Consecutive_Successes | [
"Expectation"
] | [
"Definition:Positive/Integer",
"Definition:Experiment",
"Definition:Probability",
"Definition:Experiment",
"Definition:Expectation",
"Definition:Experiment"
] | [
"Definition:Expectation",
"Definition:Experiment",
"Definition:Recursive Sequence/Recurrence Relation",
"Definition:Experiment",
"Definition:Expectation",
"Definition:Experiment",
"Definition:Probability",
"Definition:Experiment",
"Definition:Experiment",
"Definition:Expectation",
"Definition:Pr... |
proofwiki-23572 | Probability that Three Dice are Rolled in Strictly Increasing Order | Roll three fair six-sided dice one by one, recording the upwards face each time.
The probability that the dice were rolled with upwards face in strictly increasing order is $5/54$. | Let $X_1, X_2, X_3$ be the values of the three rolls.
Let $E$ be the event that:
:$X_1 < X_2 < X_3$
Let $E'$ be the event that:
:$X_1$, $X_2$ and $X_3$ are all distinct.
By Bayes' Theorem, we have:
:$\map \Pr E = \map \Pr {E \cap E'} = \map \Pr {E \mid E'} \map \Pr {E'}$
since $E' \subseteq E$.
Given that $X_1$, $X_2$... | Roll three [[Definition:Fair Die|fair]] [[Definition:Dice|six-sided dice]] one by one, recording the [[Definition:Polyhedron/Face|upwards face]] each time.
The [[Definition:Probability|probability]] that the [[Definition:Dice|dice]] were rolled with [[Definition:Polyhedron/Face|upwards face]] in [[Definition:Strictl... | Let $X_1, X_2, X_3$ be the values of the three rolls.
Let $E$ be the [[Definition:Event|event]] that:
:$X_1 < X_2 < X_3$
Let $E'$ be the [[Definition:Event|event]] that:
:$X_1$, $X_2$ and $X_3$ are all distinct.
By [[Bayes' Theorem]], we have:
:$\map \Pr E = \map \Pr {E \cap E'} = \map \Pr {E \mid E'} \map \Pr {E'}... | Probability that Three Dice are Rolled in Strictly Increasing Order | https://proofwiki.org/wiki/Probability_that_Three_Dice_are_Rolled_in_Strictly_Increasing_Order | https://proofwiki.org/wiki/Probability_that_Three_Dice_are_Rolled_in_Strictly_Increasing_Order | [
"Probability",
"Dice"
] | [
"Definition:Fair/Die",
"Definition:Die",
"Definition:Polyhedron/Face",
"Definition:Probability",
"Definition:Die",
"Definition:Polyhedron/Face",
"Definition:Strictly Increasing/Sequence"
] | [
"Definition:Event",
"Definition:Event",
"Bayes' Theorem",
"Definition:Permutation",
"Number of Permutations of All Elements",
"Definition:Independent Events",
"Bayes' Theorem",
"Definition:Probability",
"Definition:Probability",
"Category:Probability",
"Category:Dice"
] |
proofwiki-23573 | Similar Matrices have Same Eigenvalues | Let $\mathbf A$ and $\mathbf B$ be similar matrices.
Then, every eigenvalue of $\mathbf A$ is an eigenvalue of $\mathbf B$ and vice versa. | Follows directly from Similar Matrices have Same Characteristic Polynomial.
{{qed}}
Category:Matrix Similarity
Category:Eigenvalues of Square Matrices
ssyzamawrcyxcndurnqnrs02l2u8n3x | Let $\mathbf A$ and $\mathbf B$ be [[Definition:Matrix Similarity|similar matrices]].
Then, every [[Definition:Eigenvalue|eigenvalue]] of $\mathbf A$ is an [[Definition:Eigenvalue|eigenvalue]] of $\mathbf B$ and vice versa. | Follows directly from [[Similar Matrices have Same Characteristic Polynomial]].
{{qed}}
[[Category:Matrix Similarity]]
[[Category:Eigenvalues of Square Matrices]]
ssyzamawrcyxcndurnqnrs02l2u8n3x | Similar Matrices have Same Eigenvalues | https://proofwiki.org/wiki/Similar_Matrices_have_Same_Eigenvalues | https://proofwiki.org/wiki/Similar_Matrices_have_Same_Eigenvalues | [
"Matrix Similarity",
"Eigenvalues of Square Matrices"
] | [
"Definition:Matrix Similarity",
"Definition:Eigenvalue",
"Definition:Eigenvalue"
] | [
"Similar Matrices have Same Characteristic Polynomial",
"Category:Matrix Similarity",
"Category:Eigenvalues of Square Matrices"
] |
proofwiki-23574 | Euler Phi Function of Power Minus One is Divisible by Exponent | Let $a, n \in \Z$ with $a \ge 2$ and $n \ge 1$.
Then:
:$n \divides \map \phi {a^n - 1}$
where $\phi$ is the Euler phi function. | Clearly:
:$a^n = \paren {a^n - 1} + 1$
By definition of congruence:
:$a^n \equiv 1 \pmod {a^n - 1}$
From Power Function on Base Greater than One is Strictly Increasing:
:$\forall m \in \Z: 0 < m < n \implies 1 < a^m < a^n$
So $n$ is the multiplicative order of $a$ modulo $a^n - 1$.
The result follows from {{Corollary|I... | Let $a, n \in \Z$ with $a \ge 2$ and $n \ge 1$.
Then:
:$n \divides \map \phi {a^n - 1}$
where $\phi$ is the [[Definition:Euler Phi Function|Euler phi function]]. | Clearly:
:$a^n = \paren {a^n - 1} + 1$
By definition of [[Definition:Congruence (Number Theory)|congruence]]:
:$a^n \equiv 1 \pmod {a^n - 1}$
From [[Power Function on Base Greater than One is Strictly Increasing/Positive Integer|Power Function on Base Greater than One is Strictly Increasing]]:
:$\forall m \in \Z: 0 <... | Euler Phi Function of Power Minus One is Divisible by Exponent | https://proofwiki.org/wiki/Euler_Phi_Function_of_Power_Minus_One_is_Divisible_by_Exponent | https://proofwiki.org/wiki/Euler_Phi_Function_of_Power_Minus_One_is_Divisible_by_Exponent | [
"Euler Phi Function"
] | [
"Definition:Euler Phi Function"
] | [
"Definition:Congruence (Number Theory)",
"Power Function on Base Greater than One is Strictly Increasing/Positive Integer",
"Definition:Multiplicative Order of Integer",
"Category:Euler Phi Function"
] |
proofwiki-23575 | Determinant Equals Product of Eigenvalues | Let $\mathbf M$ be a square matrix over $\C$ of order $n$.
Let $\tuple {\lambda_1, \lambda_2, \ldots, \lambda_n}$ be the eigenvalues of $\mathbf M$ including algebraic multiplicity.
Then:
:$\ds \map \det {\mathbf M} = \prod_{i \mathop = 1}^n \lambda_i$ | The eigenvalues of $\mathbf M$ are the roots of its characteristic equation.
Thus:
{{begin-eqn}}
{{eqn | l = \map \det {\lambda \mathbf I - \mathbf M}
| r = \prod_{i \mathop = 1}^n \paren {\lambda - \lambda_i}
| c = Polynomial Factor Theorem
}}
{{eqn | ll= \leadsto
| l = \map \det {-\mathbf M}
|... | Let $\mathbf M$ be a [[Definition:Square Matrix|square matrix]] over $\C$ of [[Definition:Order of Matrix|order]] $n$.
Let $\tuple {\lambda_1, \lambda_2, \ldots, \lambda_n}$ be the [[Definition:Eigenvalue|eigenvalues]] of $\mathbf M$ including [[Definition:Algebraic Multiplicity|algebraic multiplicity]].
Then:
:$\ds... | The [[Definition:Eigenvalue|eigenvalues]] of $\mathbf M$ are the [[Definition:Root of Polynomial|roots]] of its [[Definition:Characteristic Equation of Matrix|characteristic equation]].
Thus:
{{begin-eqn}}
{{eqn | l = \map \det {\lambda \mathbf I - \mathbf M}
| r = \prod_{i \mathop = 1}^n \paren {\lambda - \lamb... | Determinant Equals Product of Eigenvalues/Proof 2 | https://proofwiki.org/wiki/Determinant_Equals_Product_of_Eigenvalues | https://proofwiki.org/wiki/Determinant_Equals_Product_of_Eigenvalues/Proof_2 | [
"Determinant Equals Product of Eigenvalues",
"Determinants",
"Eigenvalues of Square Matrices"
] | [
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Order",
"Definition:Eigenvalue",
"Definition:Algebraic Multiplicity"
] | [
"Definition:Eigenvalue",
"Definition:Root of Polynomial",
"Definition:Characteristic Equation of Matrix",
"Polynomial Factor Theorem",
"Determinant of Matrix Multiplied by Constant"
] |
proofwiki-23576 | Similar Matrices have Same Rank | Let $\mathbf A$ and $\mathbf B$ be similar square matrices of order $n$ over a field $\mathbb F$.
Then:
:$\map \rho {\mathbf A} = \map \rho {\mathbf B}$
where
:$\map \rho {\mathbf A}$ is the rank of $\mathbf A$ | === Full Rank Case ===
Let $\mathbf A$ be full rank:
:$\map \rho {\mathbf A} = n$
Then, by Square Matrix has Full Rank iff Nonsingular, $\mathbf A$ is nonsingular.
Therefore:
:$\map \det {\mathbf A} \neq 0$
From Similar Matrices have Same Determinant:
:$\map \det {\mathbf A} = \map \det {\mathbf B}$
Thus:
:$\map \det {... | Let $\mathbf A$ and $\mathbf B$ be [[Definition:Matrix Similarity|similar]] [[Definition:Square Matrix|square matrices]] of [[Definition:Order of Matrix|order]] $n$ over a [[Definition:Field (Abstract Algebra)|field]] $\mathbb F$.
Then:
:$\map \rho {\mathbf A} = \map \rho {\mathbf B}$
where
:$\map \rho {\mathbf A}$ i... | === Full Rank Case ===
Let $\mathbf A$ be [[Definition:Full Rank|full rank]]:
:$\map \rho {\mathbf A} = n$
Then, by [[Square Matrix has Full Rank iff Nonsingular]], $\mathbf A$ is [[Definition:Nonsingular Matrix|nonsingular]].
Therefore:
:$\map \det {\mathbf A} \neq 0$
From [[Similar Matrices have Same Determinan... | Similar Matrices have Same Rank | https://proofwiki.org/wiki/Similar_Matrices_have_Same_Rank | https://proofwiki.org/wiki/Similar_Matrices_have_Same_Rank | [
"Rank of Matrix",
"Matrix Similarity"
] | [
"Definition:Matrix Similarity",
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Order",
"Definition:Field (Abstract Algebra)",
"Definition:Rank/Matrix"
] | [
"Definition:Full Rank",
"Square Matrix has Full Rank iff Nonsingular",
"Definition:Nonsingular Matrix",
"Similar Matrices have Same Determinant",
"Definition:Nonsingular Matrix",
"Definition:Full Rank",
"Definition:Nonsingular Matrix"
] |
proofwiki-23577 | Determinant of Matrix Multiplied by Constant | Let $\mathbf M$ be a square matrix of order $n$ over a field $\mathbb F$.
Let $k$ be a scalar in $\mathbb F$.
Then:
:$\map \det {k \mathbf M} = k^n \map \det {\mathbf M}$ | For $1 \le i \le n$, let $e_i$ be the elementary row operation that multiplies row $i$ of $\mathbf A$ by $k$.
By definition of the scalar product, $k \mathbf A$ is obtained by multiplying every row of $\mathbf A$ by $k$.
That is the same as applying $e_i$ to $\mathbf A$ for each of $i \in \set {1, 2, \ldots, n}$.
Let $... | Let $\mathbf M$ be a [[Definition:Square Matrix|square matrix]] of [[Definition:Order of Matrix|order]] $n$ over a [[Definition:Field (Abstract Algebra)|field]] $\mathbb F$.
Let $k$ be a [[Definition:Scalar|scalar]] in $\mathbb F$.
Then:
:$\map \det {k \mathbf M} = k^n \map \det {\mathbf M}$ | For $1 \le i \le n$, let $e_i$ be the [[Definition:Elementary Row Operation|elementary row operation]] that [[Definition:Matrix Scalar Product|multiplies]] [[Definition:Row of Matrix|row]] $i$ of $\mathbf A$ by $k$.
By definition of the [[Definition:Matrix Scalar Product|scalar product]], $k \mathbf A$ is obtained by ... | Determinant of Matrix Multiplied by Constant/Proof 3 | https://proofwiki.org/wiki/Determinant_of_Matrix_Multiplied_by_Constant | https://proofwiki.org/wiki/Determinant_of_Matrix_Multiplied_by_Constant/Proof_3 | [
"Determinant of Matrix Multiplied by Constant",
"Determinants"
] | [
"Definition:Matrix/Square Matrix",
"Definition:Matrix/Order",
"Definition:Field (Abstract Algebra)",
"Definition:Scalar"
] | [
"Definition:Elementary Operation/Row",
"Definition:Matrix Scalar Product",
"Definition:Matrix/Row",
"Definition:Matrix Scalar Product",
"Definition:Matrix Scalar Product",
"Definition:Matrix/Row",
"Definition:Elementary Matrix/Row Operation",
"Determinant of Elementary Row Matrix/Scale Row",
"Determ... |
proofwiki-23578 | Rule of Transposition/Formulation 2/Forward Implication | :$\vdash \paren {p \implies q} \implies \paren {\neg q \implies \neg p}$ | {{BeginTableau |\vdash \paren {p \implies q} \implies \paren {\neg q \implies \neg p} }}
{{Assumption |1|p \implies q}}
{{Assumption |2|\neg q}}
{{ModusTollens |3|1, 2|\neg p|1|2}}
{{Implication |4|1|\neg q \implies \neg p|2|3}}
{{Implication |5||\paren {p \implies q} \implies \paren {\neg q \implies \neg p}|1|4}... | :$\vdash \paren {p \implies q} \implies \paren {\neg q \implies \neg p}$ | {{BeginTableau |\vdash \paren {p \implies q} \implies \paren {\neg q \implies \neg p} }}
{{Assumption |1|p \implies q}}
{{Assumption |2|\neg q}}
{{ModusTollens |3|1, 2|\neg p|1|2}}
{{Implication |4|1|\neg q \implies \neg p|2|3}}
{{Implication |5||\paren {p \implies q} \implies \paren {\neg q \implies \neg p}|1|4}... | Rule of Transposition/Formulation 2/Forward Implication/Proof 1 | https://proofwiki.org/wiki/Rule_of_Transposition/Formulation_2/Forward_Implication | https://proofwiki.org/wiki/Rule_of_Transposition/Formulation_2/Forward_Implication/Proof_1 | [
"Rule of Transposition"
] | [] | [] |
proofwiki-23579 | Rule of Transposition/Formulation 2/Forward Implication | :$\vdash \paren {p \implies q} \implies \paren {\neg q \implies \neg p}$ | We apply the Method of Truth Tables to the proposition.
As can be seen by inspection, the truth value under the main connective is true for all boolean interpretations.
:<nowiki>$\begin {array} {|ccc|c|ccccc|} \hline
(p & \implies & q) & \implies & (\neg & q & \implies & \neg & p) \\
\hline
\F & \T & \F & \T & \T & \F ... | :$\vdash \paren {p \implies q} \implies \paren {\neg q \implies \neg p}$ | We apply the [[Method of Truth Tables]] to the proposition.
As can be seen by inspection, the [[Definition:Truth Value|truth value]] under the [[Definition:Main Connective (Propositional Logic)|main connective]] is [[Definition:True|true]] for all [[Definition:Boolean Interpretation|boolean interpretations]].
:<nowik... | Rule of Transposition/Formulation 2/Forward Implication/Proof by Truth Table | https://proofwiki.org/wiki/Rule_of_Transposition/Formulation_2/Forward_Implication | https://proofwiki.org/wiki/Rule_of_Transposition/Formulation_2/Forward_Implication/Proof_by_Truth_Table | [
"Rule of Transposition"
] | [] | [
"Method of Truth Tables",
"Definition:Truth Value",
"Definition:Main Connective/Propositional Logic",
"Definition:True",
"Definition:Boolean Interpretation"
] |
proofwiki-23580 | Rule of Transposition/Formulation 2/Reverse Implication | :$\vdash \paren {\neg q \implies \neg p} \implies \paren {p \implies q}$ | {{BeginTableau|\vdash \paren {\neg q \implies \neg p} \implies \paren {p \implies q} }}
{{Assumption|1|\neg q \implies \neg p}}
{{Assumption|2|p}}
{{DoubleNegIntro|3|2|\neg \neg p|2}}
{{ModusTollens|4|1, 2|\neg \neg q|1|3}}
{{DoubleNegElimination|5|1, 2|q|4}}
{{Implication|6|1|p \implies q|2|5}}
{{Implication|7||\paren... | :$\vdash \paren {\neg q \implies \neg p} \implies \paren {p \implies q}$ | {{BeginTableau|\vdash \paren {\neg q \implies \neg p} \implies \paren {p \implies q} }}
{{Assumption|1|\neg q \implies \neg p}}
{{Assumption|2|p}}
{{DoubleNegIntro|3|2|\neg \neg p|2}}
{{ModusTollens|4|1, 2|\neg \neg q|1|3}}
{{DoubleNegElimination|5|1, 2|q|4}}
{{Implication|6|1|p \implies q|2|5}}
{{Implication|7||\paren... | Rule of Transposition/Formulation 2/Reverse Implication/Proof 1 | https://proofwiki.org/wiki/Rule_of_Transposition/Formulation_2/Reverse_Implication | https://proofwiki.org/wiki/Rule_of_Transposition/Formulation_2/Reverse_Implication/Proof_1 | [
"Rule of Transposition"
] | [] | [] |
proofwiki-23581 | Rule of Transposition/Formulation 2/Reverse Implication | :$\vdash \paren {\neg q \implies \neg p} \implies \paren {p \implies q}$ | We apply the Method of Truth Tables to the proposition.
As can be seen by inspection, the truth value under the main connective is is true for all boolean interpretations.
:<nowiki>$\begin {array} {|ccccc|c|ccc|} \hline
(\neg & q & \implies & \neg & p) & \implies & (p & \implies & q) \\
\hline
\T & \F & \T & \T & \F & ... | :$\vdash \paren {\neg q \implies \neg p} \implies \paren {p \implies q}$ | We apply the [[Method of Truth Tables]] to the proposition.
As can be seen by inspection, the [[Definition:Truth Value|truth value]] under the [[Definition:Main Connective (Propositional Logic)|main connective]] is is [[Definition:True|true]] for all [[Definition:Boolean Interpretation|boolean interpretations]].
:<no... | Rule of Transposition/Formulation 2/Reverse Implication/Proof by Truth Table | https://proofwiki.org/wiki/Rule_of_Transposition/Formulation_2/Reverse_Implication | https://proofwiki.org/wiki/Rule_of_Transposition/Formulation_2/Reverse_Implication/Proof_by_Truth_Table | [
"Rule of Transposition"
] | [] | [
"Method of Truth Tables",
"Definition:Truth Value",
"Definition:Main Connective/Propositional Logic",
"Definition:True",
"Definition:Boolean Interpretation"
] |
proofwiki-23582 | Identity Mapping is Normal | The identity mapping is normal. | Follows directly from Identity Mapping is Hermitian and Hermitian Operator is Normal.
Category:Identity Mappings
Category:Normal Operators
cy43ogpqgzu7rhz3l3qds6u6xh3z0f8 | The [[Definition:Identity Mapping|identity mapping]] is [[Definition:Normal Operator|normal]]. | Follows directly from [[Identity Mapping is Hermitian]] and [[Hermitian Operator is Normal]].
[[Category:Identity Mappings]]
[[Category:Normal Operators]]
cy43ogpqgzu7rhz3l3qds6u6xh3z0f8 | Identity Mapping is Normal | https://proofwiki.org/wiki/Identity_Mapping_is_Normal | https://proofwiki.org/wiki/Identity_Mapping_is_Normal | [
"Identity Mappings",
"Normal Operators",
"Identity Mappings",
"Normal Operators"
] | [
"Definition:Identity Mapping",
"Definition:Normal Operator"
] | [
"Identity Mapping is Hermitian",
"Hermitian Operator is Normal",
"Category:Identity Mappings",
"Category:Normal Operators"
] |
proofwiki-23583 | Adjoint is Conjugate Linear | Let $\HH$ and $\KK$ be Hilbert spaces over $\Bbb F \in \set {\R, \C}$.
Let $\map \BB {\HH, \KK}$ be the set of bounded linear transformations from $\HH$ to $\KK$.
Let $A, B \in \map \BB {\HH, \KK}$ be bounded linear transformations.
Let $\lambda \in \mathbb F$ be a scalar.
Then:
:$\paren {A + \lambda B}^* = A^* + \bar ... | Let $x \in \HH$ and $y \in \KK$ be vectors.
We have:
{{begin-eqn}}
{{eqn | l = \innerprod x {\paren {A + \lambda B}^* y}
| r = \innerprod {\paren {A + \lambda B} x} y
| c = {{Defof|Adjoint Linear Transformation}}
}}
{{eqn | r = \innerprod {A x + \lambda B x} y
}}
{{eqn | r = \innerprod {A x} y + \lambda \in... | Let $\HH$ and $\KK$ be [[Definition:Hilbert Space|Hilbert spaces]] over $\Bbb F \in \set {\R, \C}$.
Let $\map \BB {\HH, \KK}$ be the [[Definition:Set|set]] of [[Definition:Bounded Linear Transformation|bounded linear transformations]] from $\HH$ to $\KK$.
Let $A, B \in \map \BB {\HH, \KK}$ be [[Definition:Bounded Lin... | Let $x \in \HH$ and $y \in \KK$ be [[Definition:Vector|vectors]].
We have:
{{begin-eqn}}
{{eqn | l = \innerprod x {\paren {A + \lambda B}^* y}
| r = \innerprod {\paren {A + \lambda B} x} y
| c = {{Defof|Adjoint Linear Transformation}}
}}
{{eqn | r = \innerprod {A x + \lambda B x} y
}}
{{eqn | r = \innerpro... | Adjoint is Conjugate Linear/Proof 1 | https://proofwiki.org/wiki/Adjoint_is_Conjugate_Linear | https://proofwiki.org/wiki/Adjoint_is_Conjugate_Linear/Proof_1 | [
"Adjoint is Conjugate Linear",
"Conjugate Linear Mappings",
"Adjoint Linear Transformations"
] | [
"Definition:Hilbert Space",
"Definition:Set",
"Definition:Bounded Linear Transformation",
"Definition:Bounded Linear Transformation",
"Definition:Scalar/Vector Space",
"Definition:Adjoint Linear Transformation",
"Definition:Complex Conjugate",
"Definition:Adjoint Linear Transformation",
"Definition:... | [
"Definition:Vector",
"Inner Product is Sesquilinear",
"Inner Product is Sesquilinear",
"Inner Product is Sesquilinear",
"Existence and Uniqueness of Adjoint"
] |
proofwiki-23584 | Adjoint is Conjugate Linear | Let $\HH$ and $\KK$ be Hilbert spaces over $\Bbb F \in \set {\R, \C}$.
Let $\map \BB {\HH, \KK}$ be the set of bounded linear transformations from $\HH$ to $\KK$.
Let $A, B \in \map \BB {\HH, \KK}$ be bounded linear transformations.
Let $\lambda \in \mathbb F$ be a scalar.
Then:
:$\paren {A + \lambda B}^* = A^* + \bar ... | Let $\innerprod \cdot \cdot_\HH$ and $\innerprod \cdot \cdot_\KK$ be inner products on $\HH$ and $\KK$ respectively.
=== Lemma $1$ ===
{{:Adjoint is Conjugate Linear/Lemma 1}}{{qed|lemma}}
=== Lemma $2$ ===
{{:Adjoint is Conjugate Linear/Lemma 2}}{{qed|lemma}}
Thus:
{{begin-eqn}}
{{eqn | l = \paren {A + \lambda B}^*
... | Let $\HH$ and $\KK$ be [[Definition:Hilbert Space|Hilbert spaces]] over $\Bbb F \in \set {\R, \C}$.
Let $\map \BB {\HH, \KK}$ be the [[Definition:Set|set]] of [[Definition:Bounded Linear Transformation|bounded linear transformations]] from $\HH$ to $\KK$.
Let $A, B \in \map \BB {\HH, \KK}$ be [[Definition:Bounded Lin... | Let $\innerprod \cdot \cdot_\HH$ and $\innerprod \cdot \cdot_\KK$ be [[Definition:Inner Product|inner products]] on $\HH$ and $\KK$ respectively.
=== [[Adjoint is Conjugate Linear/Lemma 1|Lemma $1$]] ===
{{:Adjoint is Conjugate Linear/Lemma 1}}{{qed|lemma}}
=== [[Adjoint is Conjugate Linear/Lemma 2|Lemma $2$]] ===
... | Adjoint is Conjugate Linear/Proof 2 | https://proofwiki.org/wiki/Adjoint_is_Conjugate_Linear | https://proofwiki.org/wiki/Adjoint_is_Conjugate_Linear/Proof_2 | [
"Adjoint is Conjugate Linear",
"Conjugate Linear Mappings",
"Adjoint Linear Transformations"
] | [
"Definition:Hilbert Space",
"Definition:Set",
"Definition:Bounded Linear Transformation",
"Definition:Bounded Linear Transformation",
"Definition:Scalar/Vector Space",
"Definition:Adjoint Linear Transformation",
"Definition:Complex Conjugate",
"Definition:Adjoint Linear Transformation",
"Definition:... | [
"Definition:Inner Product",
"Adjoint is Conjugate Linear/Lemma 1",
"Adjoint is Conjugate Linear/Lemma 2"
] |
proofwiki-23585 | Operator is Normal iff Operator Minus Multiple of Identity Operator is Normal | Let $\HH$ be a Hilbert space.
Let $\mathbf T: \HH \to \HH$ be a bounded linear operator.
Let $\lambda$ be a scalar under $\HH$.
Let $\mathbf I$ be the identity operator.
Then $\mathbf T$ is normal {{iff}} $\mathbf T - \lambda \mathbf I$ is normal. | === Sufficient Case ===
We have that $\mathbf T$ is normal.
Therefore:
{{begin-eqn}}
{{eqn | l = \paren {\mathbf T - \lambda \mathbf I} \paren {\mathbf T - \lambda \mathbf I}^*
| r = \paren {\mathbf T - \lambda \mathbf I} \paren {\mathbf T^* - \overline \lambda \mathbf I^*}
| c = Adjoint is Conjugate Linear... | Let $\HH$ be a [[Definition:Hilbert Space|Hilbert space]].
Let $\mathbf T: \HH \to \HH$ be a [[Definition:Bounded Linear Operator|bounded linear operator]].
Let $\lambda$ be a [[Definition:Scalar (Vector Space)|scalar]] under $\HH$.
Let $\mathbf I$ be the [[Definition:Identity Operator|identity operator]].
Then $\... | === Sufficient Case ===
We have that $\mathbf T$ is [[Definition:Normal Operator|normal]].
Therefore:
{{begin-eqn}}
{{eqn | l = \paren {\mathbf T - \lambda \mathbf I} \paren {\mathbf T - \lambda \mathbf I}^*
| r = \paren {\mathbf T - \lambda \mathbf I} \paren {\mathbf T^* - \overline \lambda \mathbf I^*}
... | Operator is Normal iff Operator Minus Multiple of Identity Operator is Normal | https://proofwiki.org/wiki/Operator_is_Normal_iff_Operator_Minus_Multiple_of_Identity_Operator_is_Normal | https://proofwiki.org/wiki/Operator_is_Normal_iff_Operator_Minus_Multiple_of_Identity_Operator_is_Normal | [
"Normal Operators"
] | [
"Definition:Hilbert Space",
"Definition:Bounded Linear Operator",
"Definition:Scalar/Vector Space",
"Definition:Identity Mapping",
"Definition:Normal Operator",
"Definition:Normal Operator"
] | [
"Definition:Normal Operator",
"Adjoint is Conjugate Linear",
"Identity Mapping is Normal",
"Identity Mapping is Hermitian",
"Adjoint is Conjugate Linear",
"Definition:Normal Operator",
"Definition:Normal Operator",
"Adjoint is Conjugate Linear",
"Identity Mapping is Hermitian",
"Definition:Normal ... |
proofwiki-23586 | Adjoint of Normal Operator has Same Eigenvectors and Complex Conjugated Eigenvalues | Let $\HH$ be a Hilbert space.
Let $\mathbf T: \HH \to \HH$ be a normal operator.
Let $\lambda$ be an eigenvalue of $\mathbf T$.
Let $\mathbf v$ be a eigenvector of $\mathbf T$ associated with $\lambda$.
Then, $\mathbf v$ is also an eigenvector of the adjoint $\mathbf T^*$ with an associated eigenvalue $\overline \lambd... | We have:
{{begin-eqn}}
{{eqn | l = \mathbf {T v}
| r = \lambda \mathbf v
}}
{{eqn | ll= \leadstoandfrom
| l = \mathbf 0
| r = \mathbf {T v} - \lambda \mathbf v
}}
{{eqn | ll= \leadstoandfrom
| l = 0
| r = \norm {\paren {\mathbf T - \lambda \mathbf I} \mathbf v}
| c = Norm is positive... | Let $\HH$ be a [[Definition:Hilbert Space|Hilbert space]].
Let $\mathbf T: \HH \to \HH$ be a [[Definition:Normal Operator|normal operator]].
Let $\lambda$ be an [[Definition:Eigenvalue of Linear Operator|eigenvalue]] of $\mathbf T$.
Let $\mathbf v$ be a [[Definition:Eigenvector of Linear Operator|eigenvector]] of $\... | We have:
{{begin-eqn}}
{{eqn | l = \mathbf {T v}
| r = \lambda \mathbf v
}}
{{eqn | ll= \leadstoandfrom
| l = \mathbf 0
| r = \mathbf {T v} - \lambda \mathbf v
}}
{{eqn | ll= \leadstoandfrom
| l = 0
| r = \norm {\paren {\mathbf T - \lambda \mathbf I} \mathbf v}
| c = [[Definition:Nor... | Adjoint of Normal Operator has Same Eigenvectors and Complex Conjugated Eigenvalues | https://proofwiki.org/wiki/Adjoint_of_Normal_Operator_has_Same_Eigenvectors_and_Complex_Conjugated_Eigenvalues | https://proofwiki.org/wiki/Adjoint_of_Normal_Operator_has_Same_Eigenvectors_and_Complex_Conjugated_Eigenvalues | [
"Normal Operators",
"Eigenvectors of Linear Operators",
"Eigenvalues of Linear Operators",
"Adjoints"
] | [
"Definition:Hilbert Space",
"Definition:Normal Operator",
"Definition:Eigenvalue/Linear Operator",
"Definition:Eigenvector/Linear Operator",
"Definition:Eigenvector/Linear Operator",
"Definition:Adjoint Linear Transformation",
"Definition:Eigenvalue/Linear Operator",
"Definition:Complex Conjugate"
] | [
"Definition:Norm/Vector Space",
"Definition:Positive Definite (Ring)",
"Operator is Normal iff Operator Minus Multiple of Identity Operator is Normal",
"Characterization of Normal Operators",
"Adjoint is Conjugate Linear",
"Definition:Norm/Vector Space",
"Definition:Positive Definite (Ring)",
"Definit... |
proofwiki-23587 | Eigenvalues of Linear Operator Shifted by Scaled Identity Operator | Let $V$ be a vector space over a field $\mathbb F$.
Let $A : V \to V$ be a linear operator.
Let $s$ be a scalar in $\mathbb F$.
Let $I$ be the identity operator on $V$.
Then, the operator $T + s I$ has the same eigenvectors as $A$ with all eigenvalues shifted by $s$. | Let $\lambda$ be an eigenvalue of $A$.
Let $v$ be an eigenvector of $A$ associated with $\lambda$.
We have:
{{begin-eqn}}
{{eqn | l = \paren {T + s I} v
| r = T v + s v
}}
{{eqn | r = \lambda v + s v
}}
{{eqn | r = \paren {\lambda + s} v
}}
{{end-eqn}}
Therefore, $v$ is an eigenvector of $T + s I$ with a shifted ... | Let $V$ be a [[Definition:Vector Space|vector space]] over a [[Definition:Field (Abstract Algebra)|field]] $\mathbb F$.
Let $A : V \to V$ be a [[Definition:Linear Operator|linear operator]].
Let $s$ be a [[Definition:Scalar (Vector Space)|scalar]] in $\mathbb F$.
Let $I$ be the [[Definition:Identity Operator|identit... | Let $\lambda$ be an [[Definition:Eigenvalue of Linear Operator|eigenvalue]] of $A$.
Let $v$ be an [[Definition:Eigenvector of Linear Operator|eigenvector]] of $A$ associated with $\lambda$.
We have:
{{begin-eqn}}
{{eqn | l = \paren {T + s I} v
| r = T v + s v
}}
{{eqn | r = \lambda v + s v
}}
{{eqn | r = \pare... | Eigenvalues of Linear Operator Shifted by Scaled Identity Operator | https://proofwiki.org/wiki/Eigenvalues_of_Linear_Operator_Shifted_by_Scaled_Identity_Operator | https://proofwiki.org/wiki/Eigenvalues_of_Linear_Operator_Shifted_by_Scaled_Identity_Operator | [
"Linear Operators",
"Eigenvalues of Linear Operators",
"Identity Mappings"
] | [
"Definition:Vector Space",
"Definition:Field (Abstract Algebra)",
"Definition:Linear Operator",
"Definition:Scalar/Vector Space",
"Definition:Identity Mapping",
"Definition:Linear Operator",
"Definition:Eigenvector/Linear Operator",
"Definition:Eigenvalue/Linear Operator"
] | [
"Definition:Eigenvalue/Linear Operator",
"Definition:Eigenvector/Linear Operator",
"Definition:Eigenvector/Linear Operator",
"Definition:Eigenvalue/Linear Operator",
"Category:Linear Operators",
"Category:Eigenvalues of Linear Operators",
"Category:Identity Mappings"
] |
proofwiki-23588 | Eckmann-Hilton Argument | Let $S$ be a non-empty set.
Let $\circ$ and $\ast$ be binary operations on $S$ such that:
{{begin-itemize}}
{{item|(\text a):|$\circ$ and $\ast$ have respective identity elements $e_\circ$ and $e_\ast$ in $S$}}
{{item|(\text b):|$\forall a, b, c, d \in S: \paren {a \circ b} \ast \paren {c \circ d} {{=}} \paren {a \ast ... | {{begin-eqn}}
{{eqn | l = e_\circ
| r = e_\circ \circ e_\circ
| c = {{Defof|Identity Element}}: $e_\circ$
}}
{{eqn | r = \paren {e_\circ \ast e_\ast} \circ \paren {e_\ast \ast e_\circ}
| c = {{Defof|Identity Element}}: $e_\ast$
}}
{{eqn | r = \paren {e_\circ \circ e_\ast} \ast \paren {e_\ast \circ e_\... | Let $S$ be a [[Definition:Non-Empty Set|non-empty set]].
Let $\circ$ and $\ast$ be [[Definition:Binary Operation|binary operations]] on $S$ such that:
{{begin-itemize}}
{{item|(\text a):|$\circ$ and $\ast$ have respective [[Definition:Identity Element|identity elements]] $e_\circ$ and $e_\ast$ in $S$}}
{{item|(\text b... | {{begin-eqn}}
{{eqn | l = e_\circ
| r = e_\circ \circ e_\circ
| c = {{Defof|Identity Element}}: $e_\circ$
}}
{{eqn | r = \paren {e_\circ \ast e_\ast} \circ \paren {e_\ast \ast e_\circ}
| c = {{Defof|Identity Element}}: $e_\ast$
}}
{{eqn | r = \paren {e_\circ \circ e_\ast} \ast \paren {e_\ast \circ e_\... | Eckmann-Hilton Argument | https://proofwiki.org/wiki/Eckmann-Hilton_Argument | https://proofwiki.org/wiki/Eckmann-Hilton_Argument | [
"Commutative Monoids"
] | [
"Definition:Non-Empty Set",
"Definition:Operation/Binary Operation",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Commutative/Operation"
] | [
"Equality of Mappings",
"Definition:Commutative/Operation"
] |
proofwiki-23589 | Peirce's Law/Strong Form/Formulation 1 | :$\paren {\paren {p \implies q} \implies p} \dashv \vdash p$ | === $(1):$ $\vdash$ Direction ===
{{:Peirce's Law/Strong Form/Formulation 1/Forward Direction}} | :$\paren {\paren {p \implies q} \implies p} \dashv \vdash p$ | === $(1):$ [[Peirce's Law/Strong Form/Formulation 1/Forward Direction|$\vdash$ Direction]] ===
{{:Peirce's Law/Strong Form/Formulation 1/Forward Direction}} | Peirce's Law/Strong Form/Formulation 1 | https://proofwiki.org/wiki/Peirce's_Law/Strong_Form/Formulation_1 | https://proofwiki.org/wiki/Peirce's_Law/Strong_Form/Formulation_1 | [
"Peirce's Law"
] | [] | [
"Peirce's Law/Strong Form/Formulation 1/Forward Direction"
] |
proofwiki-23590 | Peirce's Law/Strong Form/Formulation 1 | :$\paren {\paren {p \implies q} \implies p} \dashv \vdash p$ | We apply the Method of Truth Tables.
As can be seen by inspection, the truth values under the main connectives match for all boolean interpretations.
:<nowiki>$\begin{array}{|ccccc||c|}\hline
((p & \implies & q) & \implies & p) & p \\
\hline
\F & \T & \F & \F & \F & \F \\
\F & \T & \T & \F & \F & \F \\
\T & \F & \F & \... | :$\paren {\paren {p \implies q} \implies p} \dashv \vdash p$ | We apply the [[Method of Truth Tables]].
As can be seen by inspection, the [[Definition:Truth Value|truth values]] under the [[Definition:Main Connective (Propositional Logic)|main connectives]] match for all [[Definition:Boolean Interpretation|boolean interpretations]].
:<nowiki>$\begin{array}{|ccccc||c|}\hline
((p ... | Peirce's Law/Strong Form/Formulation 1/Proof by Truth Table | https://proofwiki.org/wiki/Peirce's_Law/Strong_Form/Formulation_1 | https://proofwiki.org/wiki/Peirce's_Law/Strong_Form/Formulation_1/Proof_by_Truth_Table | [
"Peirce's Law"
] | [] | [
"Method of Truth Tables",
"Definition:Truth Value",
"Definition:Main Connective/Propositional Logic",
"Definition:Boolean Interpretation"
] |
proofwiki-23591 | Peirce's Law/Strong Form/Formulation 1/Forward Direction | :$\paren {\paren {p \implies q} \implies p} \vdash p$ | This is none other than Peirce's law itself, whose proof we transclude here:
{{:Peirce's Law/Formulation 1/Proof 2}} | :$\paren {\paren {p \implies q} \implies p} \vdash p$ | This is none other than [[Peirce's Law/Formulation 1|Peirce's law]] itself, whose [[Peirce's Law/Formulation 1/Proof 2|proof]] we transclude here:
{{:Peirce's Law/Formulation 1/Proof 2}} | Peirce's Law/Strong Form/Formulation 1/Forward Direction | https://proofwiki.org/wiki/Peirce's_Law/Strong_Form/Formulation_1/Forward_Direction | https://proofwiki.org/wiki/Peirce's_Law/Strong_Form/Formulation_1/Forward_Direction | [
"Peirce's Law"
] | [] | [
"Peirce's Law/Formulation 1",
"Peirce's Law/Formulation 1/Proof 2"
] |
proofwiki-23592 | Peirce's Law/Strong Form/Formulation 1/Reverse Direction | :$\paren {\paren {p \implies q} \implies p} \dashv p$ | {{BeginTableau|p \vdash \paren {p \implies q} \implies p}}
{{Premise|1|p}}
{{SequentIntro|2|1|\paren {p \implies q} \implies p|1|True Statement is implied by Every Statement}}
{{EndTableau|qed}} | :$\paren {\paren {p \implies q} \implies p} \dashv p$ | {{BeginTableau|p \vdash \paren {p \implies q} \implies p}}
{{Premise|1|p}}
{{SequentIntro|2|1|\paren {p \implies q} \implies p|1|[[True Statement is implied by Every Statement]]}}
{{EndTableau|qed}} | Peirce's Law/Strong Form/Formulation 1/Reverse Direction | https://proofwiki.org/wiki/Peirce's_Law/Strong_Form/Formulation_1/Reverse_Direction | https://proofwiki.org/wiki/Peirce's_Law/Strong_Form/Formulation_1/Reverse_Direction | [
"Peirce's Law",
"Peirce's Law"
] | [] | [
"True Statement is implied by Every Statement"
] |
proofwiki-23593 | Peirce's Law/Strong Form/Formulation 2/Forward Direction | :$\paren {\paren {p \implies q} \implies p} \implies p$ | This is none other than Peirce's law itself, whose proof we transclude here:
{{:Peirce's Law/Formulation 2/Proof 1}} | :$\paren {\paren {p \implies q} \implies p} \implies p$ | This is none other than [[Peirce's Law/Formulation 2|Peirce's law]] itself, whose [[Peirce's Law/Formulation 2/Proof 1|proof]] we transclude here:
{{:Peirce's Law/Formulation 2/Proof 1}} | Peirce's Law/Strong Form/Formulation 2/Forward Direction | https://proofwiki.org/wiki/Peirce's_Law/Strong_Form/Formulation_2/Forward_Direction | https://proofwiki.org/wiki/Peirce's_Law/Strong_Form/Formulation_2/Forward_Direction | [
"Peirce's Law",
"Peirce's Law"
] | [] | [
"Peirce's Law/Formulation 2",
"Peirce's Law/Formulation 2/Proof 1"
] |
proofwiki-23594 | Peirce's Law/Strong Form/Formulation 2/Reverse Direction | :$\vdash p \implies \paren {\paren {p \implies q} \implies p}$ | {{BeginTableau|\vdash p \implies \paren {\paren {p \implies q} \implies p} }}
{{Assumption|1|p}}
{{SequentIntro|2|1|\paren {p \implies q} \implies p|1|Peirce's Law Strong Form: Formulation 1: $p \vdash \paren {p \implies q} \implies p$}}
{{Implication|3||p \implies \paren {\paren {p \implies q} \implies p}|1|2}}
{{EndT... | :$\vdash p \implies \paren {\paren {p \implies q} \implies p}$ | {{BeginTableau|\vdash p \implies \paren {\paren {p \implies q} \implies p} }}
{{Assumption|1|p}}
{{SequentIntro|2|1|\paren {p \implies q} \implies p|1|[[Peirce's Law/Strong Form/Formulation 1/Reverse Direction|Peirce's Law Strong Form: Formulation 1]]: $p \vdash \paren {p \implies q} \implies p$}}
{{Implication|3||p \i... | Peirce's Law/Strong Form/Formulation 2/Reverse Direction | https://proofwiki.org/wiki/Peirce's_Law/Strong_Form/Formulation_2/Reverse_Direction | https://proofwiki.org/wiki/Peirce's_Law/Strong_Form/Formulation_2/Reverse_Direction | [
"Peirce's Law"
] | [] | [
"Peirce's Law/Strong Form/Formulation 1/Reverse Direction"
] |
proofwiki-23595 | (p implies q) implies q, q implies p therefore p | :$\paren {p \implies q} \implies q, q \implies p \vdash p$ | {{BeginTableau|\paren {p \implies q} \implies q, q \implies p \vdash p}}
{{Premise|1|\paren {p \implies q} \implies q}}
{{Premise|2|q \implies p}}
{{Assumption|3|\neg p}}
{{ModusTollens|4|2,3|\neg q|2|3}}
{{ModusTollens|5|1,2,3|\neg \paren {p \implies q}|1|4}}
{{SequentIntro|6|3|p \implies q|3|False Statement implies E... | :$\paren {p \implies q} \implies q, q \implies p \vdash p$ | {{BeginTableau|\paren {p \implies q} \implies q, q \implies p \vdash p}}
{{Premise|1|\paren {p \implies q} \implies q}}
{{Premise|2|q \implies p}}
{{Assumption|3|\neg p}}
{{ModusTollens|4|2,3|\neg q|2|3}}
{{ModusTollens|5|1,2,3|\neg \paren {p \implies q}|1|4}}
{{SequentIntro|6|3|p \implies q|3|[[False Statement implies... | (p implies q) implies q, q implies p therefore p/Proof 1 | https://proofwiki.org/wiki/(p_implies_q)_implies_q,_q_implies_p_therefore_p | https://proofwiki.org/wiki/(p_implies_q)_implies_q,_q_implies_p_therefore_p/Proof_1 | [
"(p implies q) implies q, q implies p therefore p",
"Conditional"
] | [] | [
"False Statement implies Every Statement"
] |
proofwiki-23596 | (p implies q) implies q, q implies p therefore p | :$\paren {p \implies q} \implies q, q \implies p \vdash p$ | {{BeginTableau|\paren {p \implies q} \implies q, q \implies p \vdash p}}
{{Premise|1|\paren {p \implies q} \implies q}}
{{Premise|2|q \implies p}}
{{Assumption|3|\neg p}}
{{ModusTollens|4|2, 3|\neg q|2|3}}
{{ModusTollens|5|1, 2, 3|\neg \paren {p \implies q}|1|4}}
{{Assumption|6|p}}
{{NonContradiction|7|3, 6|3|6}}
{{Exp... | :$\paren {p \implies q} \implies q, q \implies p \vdash p$ | {{BeginTableau|\paren {p \implies q} \implies q, q \implies p \vdash p}}
{{Premise|1|\paren {p \implies q} \implies q}}
{{Premise|2|q \implies p}}
{{Assumption|3|\neg p}}
{{ModusTollens|4|2, 3|\neg q|2|3}}
{{ModusTollens|5|1, 2, 3|\neg \paren {p \implies q}|1|4}}
{{Assumption|6|p}}
{{NonContradiction|7|3, 6|3|6}}
{{Exp... | (p implies q) implies q, q implies p therefore p/Proof 2 | https://proofwiki.org/wiki/(p_implies_q)_implies_q,_q_implies_p_therefore_p | https://proofwiki.org/wiki/(p_implies_q)_implies_q,_q_implies_p_therefore_p/Proof_2 | [
"(p implies q) implies q, q implies p therefore p",
"Conditional"
] | [] | [] |
proofwiki-23597 | Simple Spectrum Approximation of Diagonalizable Matrix/Property 1 | :$\ds \lim_{t \mathop \to 0} \map {\mathbf A} t {{=}} \mathbf A$ | {{begin-eqn}}
{{eqn | l = \lim_{t \mathop \to 0} \map {\mathbf A} t
| r = \map {\mathbf A} 0
}}
{{eqn | r = \mathbf A + 0 \mathbf P \mathbf E \mathbf P^{-1}
}}
{{eqn | r = \mathbf A
}}
{{end-eqn}}
{{qed}}
Category:Simple Spectrum Approximation of Diagonalizable Matrix
8op46iqjpu9w5ns2yycw7hlh6uevfs9 | :$\ds \lim_{t \mathop \to 0} \map {\mathbf A} t {{=}} \mathbf A$ | {{begin-eqn}}
{{eqn | l = \lim_{t \mathop \to 0} \map {\mathbf A} t
| r = \map {\mathbf A} 0
}}
{{eqn | r = \mathbf A + 0 \mathbf P \mathbf E \mathbf P^{-1}
}}
{{eqn | r = \mathbf A
}}
{{end-eqn}}
{{qed}}
[[Category:Simple Spectrum Approximation of Diagonalizable Matrix]]
8op46iqjpu9w5ns2yycw7hlh6uevfs9 | Simple Spectrum Approximation of Diagonalizable Matrix/Property 1 | https://proofwiki.org/wiki/Simple_Spectrum_Approximation_of_Diagonalizable_Matrix/Property_1 | https://proofwiki.org/wiki/Simple_Spectrum_Approximation_of_Diagonalizable_Matrix/Property_1 | [
"Simple Spectrum Approximation of Diagonalizable Matrix"
] | [] | [
"Category:Simple Spectrum Approximation of Diagonalizable Matrix"
] |
proofwiki-23598 | Simple Spectrum Approximation of Diagonalizable Matrix/Property 2 | :$\map {\mathbf A} t$ is diagonalizable for all values of $t$ | Let $\mathbf D = \mathbf P^{-1} \mathbf A \mathbf P$ be $\mathbf A$ diagonalized.
{{begin-eqn}}
{{eqn | l = \map {\mathbf A} t
| r = \mathbf A + t \mathbf P \mathbf E \mathbf P^{-1}
}}
{{eqn | r = \mathbf P \mathbf D \mathbf P^{-1} + t \mathbf P \mathbf E \mathbf P^{-1}
}}
{{eqn | r = \mathbf P \paren {\mathbf D ... | :$\map {\mathbf A} t$ is [[Definition:Diagonalizable Matrix|diagonalizable]] for all values of $t$ | Let $\mathbf D = \mathbf P^{-1} \mathbf A \mathbf P$ be $\mathbf A$ [[Definition:Diagonalizable Matrix|diagonalized]].
{{begin-eqn}}
{{eqn | l = \map {\mathbf A} t
| r = \mathbf A + t \mathbf P \mathbf E \mathbf P^{-1}
}}
{{eqn | r = \mathbf P \mathbf D \mathbf P^{-1} + t \mathbf P \mathbf E \mathbf P^{-1}
}}
{{... | Simple Spectrum Approximation of Diagonalizable Matrix/Property 2 | https://proofwiki.org/wiki/Simple_Spectrum_Approximation_of_Diagonalizable_Matrix/Property_2 | https://proofwiki.org/wiki/Simple_Spectrum_Approximation_of_Diagonalizable_Matrix/Property_2 | [
"Simple Spectrum Approximation of Diagonalizable Matrix"
] | [
"Definition:Diagonalizable Matrix"
] | [
"Definition:Diagonalizable Matrix",
"Definition:Diagonal Matrix",
"Definition:Diagonal Matrix",
"Definition:Diagonalizable Matrix",
"Definition:Matrix Similarity",
"Definition:Diagonal Matrix",
"Definition:Diagonalizable Matrix",
"Category:Simple Spectrum Approximation of Diagonalizable Matrix"
] |
proofwiki-23599 | Simple Spectrum Approximation of Diagonalizable Matrix/Property 4 | :$\ds \lim_{t \mathop \to 0} \map \sigma {\map {\mathbf A} t} {{=}} \map \sigma {\mathbf A}$ | From a previous proof, we have that:
:$\map {\mathbf A} t = \mathbf P \paren {\mathbf D + t \mathbf E} \mathbf P^{-1}$
where $\mathbf D = \mathbf P^{-1} \mathbf A \mathbf P$
Let $\varepsilon_i$ be the $i^{th}$ diagonal element of $\mathbf E$.
By Characterization of Diagonalizable Matrices, the eigenvalues of $\map {\ma... | :$\ds \lim_{t \mathop \to 0} \map \sigma {\map {\mathbf A} t} {{=}} \map \sigma {\mathbf A}$ | From [[Simple Spectrum Approximation of Diagonalizable Matrix/Property 2|a previous proof]], we have that:
:$\map {\mathbf A} t = \mathbf P \paren {\mathbf D + t \mathbf E} \mathbf P^{-1}$
where $\mathbf D = \mathbf P^{-1} \mathbf A \mathbf P$
Let $\varepsilon_i$ be the $i^{th}$ [[Definition:Diagonal Element|diagonal... | Simple Spectrum Approximation of Diagonalizable Matrix/Property 4 | https://proofwiki.org/wiki/Simple_Spectrum_Approximation_of_Diagonalizable_Matrix/Property_4 | https://proofwiki.org/wiki/Simple_Spectrum_Approximation_of_Diagonalizable_Matrix/Property_4 | [
"Simple Spectrum Approximation of Diagonalizable Matrix"
] | [] | [
"Simple Spectrum Approximation of Diagonalizable Matrix/Property 2",
"Definition:Main Diagonal/Diagonal Elements",
"Characterization of Diagonalizable Matrices",
"Definition:Spectrum (Spectral Theory)/Square Matrix",
"Category:Simple Spectrum Approximation of Diagonalizable Matrix"
] |
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