statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
orthogonal_projection_orthogonal_projection_of_le {U V : submodule 𝕜 E} [complete_space U]
[complete_space V] (h : U ≤ V) (x : E) :
orthogonal_projection U (orthogonal_projection V x) = orthogonal_projection U x | eq.symm $ by simpa only [sub_eq_zero, map_sub] using
orthogonal_projection_mem_subspace_orthogonal_complement_eq_zero
(submodule.orthogonal_le h (sub_orthogonal_projection_mem_orthogonal x)) | lemma | orthogonal_projection_orthogonal_projection_of_le | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"complete_space",
"orthogonal_projection",
"orthogonal_projection_mem_subspace_orthogonal_complement_eq_zero",
"sub_orthogonal_projection_mem_orthogonal",
"submodule",
"submodule.orthogonal_le"
] | If `U ≤ V`, then projecting on `V` and then on `U` is the same as projecting on `U`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthogonal_projection_tendsto_closure_supr [complete_space E] {ι : Type*}
[semilattice_sup ι] (U : ι → submodule 𝕜 E) [∀ i, complete_space (U i)]
(hU : monotone U) (x : E) :
filter.tendsto (λ i, (orthogonal_projection (U i) x : E)) at_top
(𝓝 (orthogonal_projection (⨆ i, U i).topological_closure x : E)) | begin
casesI is_empty_or_nonempty ι,
{ rw filter_eq_bot_of_is_empty (at_top : filter ι),
exact tendsto_bot },
let y := (orthogonal_projection (⨆ i, U i).topological_closure x : E),
have proj_x : ∀ i, orthogonal_projection (U i) x = orthogonal_projection (U i) y :=
λ i, (orthogonal_projection_orthogonal_... | lemma | orthogonal_projection_tendsto_closure_supr | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"cinfi_le",
"complete_space",
"filter",
"filter.tendsto",
"is_empty_or_nonempty",
"le_supr",
"metric.mem_closure_iff",
"monotone",
"normed_add_comm_group.tendsto_at_top",
"orthogonal_projection",
"orthogonal_projection_minimal",
"orthogonal_projection_orthogonal_projection_of_le",
"semilatti... | Given a monotone family `U` of complete submodules of `E` and a fixed `x : E`,
the orthogonal projection of `x` on `U i` tends to the orthogonal projection of `x` on
`(⨆ i, U i).topological_closure` along `at_top`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthogonal_projection_tendsto_self [complete_space E] {ι : Type*} [semilattice_sup ι]
(U : ι → submodule 𝕜 E) [∀ t, complete_space (U t)] (hU : monotone U)
(x : E) (hU' : ⊤ ≤ (⨆ t, U t).topological_closure) :
filter.tendsto (λ t, (orthogonal_projection (U t) x : E)) at_top (𝓝 x) | begin
rw ← eq_top_iff at hU',
convert orthogonal_projection_tendsto_closure_supr U hU x,
rw orthogonal_projection_eq_self_iff.mpr _,
rw hU',
trivial
end | lemma | orthogonal_projection_tendsto_self | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"complete_space",
"eq_top_iff",
"filter.tendsto",
"monotone",
"orthogonal_projection",
"orthogonal_projection_tendsto_closure_supr",
"semilattice_sup",
"submodule"
] | Given a monotone family `U` of complete submodules of `E` with dense span supremum,
and a fixed `x : E`, the orthogonal projection of `x` on `U i` tends to `x` along `at_top`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
submodule.triorthogonal_eq_orthogonal [complete_space E] : Kᗮᗮᗮ = Kᗮ | begin
rw Kᗮ.orthogonal_orthogonal_eq_closure,
exact K.is_closed_orthogonal.submodule_topological_closure_eq,
end | lemma | submodule.triorthogonal_eq_orthogonal | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"complete_space"
] | The orthogonal complement satisfies `Kᗮᗮᗮ = Kᗮ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
submodule.topological_closure_eq_top_iff [complete_space E] :
K.topological_closure = ⊤ ↔ Kᗮ = ⊥ | begin
rw ←submodule.orthogonal_orthogonal_eq_closure,
split; intro h,
{ rw [←submodule.triorthogonal_eq_orthogonal, h, submodule.top_orthogonal_eq_bot] },
{ rw [h, submodule.bot_orthogonal_eq_top] }
end | lemma | submodule.topological_closure_eq_top_iff | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"complete_space",
"submodule.bot_orthogonal_eq_top",
"submodule.top_orthogonal_eq_bot"
] | The closure of `K` is the full space iff `Kᗮ` is trivial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_of_sub_mem_orthogonal (hK : dense (K : set E)) (h : x - y ∈ Kᗮ) : x = y | begin
rw [dense_iff_topological_closure_eq_top, topological_closure_eq_top_iff] at hK,
rwa [hK, submodule.mem_bot, sub_eq_zero] at h,
end | lemma | dense.eq_of_sub_mem_orthogonal | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"dense",
"submodule.mem_bot"
] | If `S` is dense and `x - y ∈ Kᗮ`, then `x = y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_zero_of_mem_orthogonal (hK : dense (K : set E)) (h : x ∈ Kᗮ) : x = 0 | hK.eq_of_sub_mem_orthogonal (by rwa [sub_zero]) | lemma | dense.eq_zero_of_mem_orthogonal | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"dense"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_of_inner_left (hK : dense (K : set E)) (h : ∀ v : K, ⟪x, v⟫ = ⟪y, v⟫) : x = y | hK.eq_of_sub_mem_orthogonal (submodule.sub_mem_orthogonal_of_inner_left h) | lemma | dense.eq_of_inner_left | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"dense",
"submodule.sub_mem_orthogonal_of_inner_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_zero_of_inner_left (hK : dense (K : set E)) (h : ∀ v : K, ⟪x, v⟫ = 0) : x = 0 | hK.eq_of_inner_left (λ v, by rw [inner_zero_left, h v]) | lemma | dense.eq_zero_of_inner_left | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"dense",
"inner_zero_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_of_inner_right (hK : dense (K : set E))
(h : ∀ v : K, ⟪(v : E), x⟫ = ⟪(v : E), y⟫) : x = y | hK.eq_of_sub_mem_orthogonal (submodule.sub_mem_orthogonal_of_inner_right h) | lemma | dense.eq_of_inner_right | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"dense",
"submodule.sub_mem_orthogonal_of_inner_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_zero_of_inner_right (hK : dense (K : set E))
(h : ∀ v : K, ⟪(v : E), x⟫ = 0) : x = 0 | hK.eq_of_inner_right (λ v, by rw [inner_zero_right, h v]) | lemma | dense.eq_zero_of_inner_right | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"dense",
"inner_zero_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
reflection_mem_subspace_orthogonal_precomplement_eq_neg
[complete_space E] {v : E} (hv : v ∈ K) :
reflection Kᗮ v = -v | reflection_mem_subspace_orthogonal_complement_eq_neg (K.le_orthogonal_orthogonal hv) | lemma | reflection_mem_subspace_orthogonal_precomplement_eq_neg | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"complete_space",
"reflection",
"reflection_mem_subspace_orthogonal_complement_eq_neg"
] | The reflection in `Kᗮ` of an element of `K` is its negation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthogonal_projection_orthogonal_complement_singleton_eq_zero [complete_space E] (v : E) :
orthogonal_projection (𝕜 ∙ v)ᗮ v = 0 | orthogonal_projection_mem_subspace_orthogonal_precomplement_eq_zero
(submodule.mem_span_singleton_self v) | lemma | orthogonal_projection_orthogonal_complement_singleton_eq_zero | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"complete_space",
"orthogonal_projection",
"orthogonal_projection_mem_subspace_orthogonal_precomplement_eq_zero",
"submodule.mem_span_singleton_self"
] | The orthogonal projection onto `(𝕜 ∙ v)ᗮ` of `v` is zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
reflection_orthogonal_complement_singleton_eq_neg [complete_space E] (v : E) :
reflection (𝕜 ∙ v)ᗮ v = -v | reflection_mem_subspace_orthogonal_precomplement_eq_neg (submodule.mem_span_singleton_self v) | lemma | reflection_orthogonal_complement_singleton_eq_neg | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"complete_space",
"reflection",
"reflection_mem_subspace_orthogonal_precomplement_eq_neg",
"submodule.mem_span_singleton_self"
] | The reflection in `(𝕜 ∙ v)ᗮ` of `v` is `-v`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
reflection_sub [complete_space F] {v w : F} (h : ‖v‖ = ‖w‖) :
reflection (ℝ ∙ (v - w))ᗮ v = w | begin
set R : F ≃ₗᵢ[ℝ] F := reflection (ℝ ∙ (v - w))ᗮ,
suffices : R v + R v = w + w,
{ apply smul_right_injective F (by norm_num : (2:ℝ) ≠ 0),
simpa [two_smul] using this },
have h₁ : R (v - w) = -(v - w) := reflection_orthogonal_complement_singleton_eq_neg (v - w),
have h₂ : R (v + w) = v + w,
{ apply ... | lemma | reflection_sub | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"complete_space",
"congr_arg2",
"real_inner_add_sub_eq_zero_iff",
"reflection",
"reflection_mem_subspace_eq_self",
"reflection_orthogonal_complement_singleton_eq_neg",
"smul_right_injective",
"submodule.mem_orthogonal_singleton_iff_inner_left",
"two_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_sum_orthogonal_projection_self_orthogonal_complement
[complete_space E] [complete_space K] (w : E) :
w = (orthogonal_projection K w : E) + (orthogonal_projection Kᗮ w : E) | begin
obtain ⟨y, hy, z, hz, hwyz⟩ := K.exists_sum_mem_mem_orthogonal w,
convert hwyz,
{ exact eq_orthogonal_projection_of_mem_orthogonal' hy hz hwyz },
{ rw add_comm at hwyz,
refine eq_orthogonal_projection_of_mem_orthogonal' hz _ hwyz,
simp [hy] }
end | lemma | eq_sum_orthogonal_projection_self_orthogonal_complement | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"complete_space",
"eq_orthogonal_projection_of_mem_orthogonal'",
"orthogonal_projection"
] | In a complete space `E`, a vector splits as the sum of its orthogonal projections onto a
complete submodule `K` and onto the orthogonal complement of `K`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_sq_eq_add_norm_sq_projection
(x : E) (S : submodule 𝕜 E) [complete_space E] [complete_space S] :
‖x‖^2 = ‖orthogonal_projection S x‖^2 + ‖orthogonal_projection Sᗮ x‖^2 | begin
let p1 := orthogonal_projection S,
let p2 := orthogonal_projection Sᗮ,
have x_decomp : x = p1 x + p2 x :=
eq_sum_orthogonal_projection_self_orthogonal_complement S x,
have x_orth : ⟪ (p1 x : E), p2 x ⟫ = 0 :=
submodule.inner_right_of_mem_orthogonal (set_like.coe_mem (p1 x)) (set_like.coe_mem (p2 x... | lemma | norm_sq_eq_add_norm_sq_projection | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"complete_space",
"eq_sum_orthogonal_projection_self_orthogonal_complement",
"mul_eq_mul_left_iff",
"norm_add_sq_eq_norm_sq_add_norm_sq_of_inner_eq_zero",
"norm_eq_zero",
"orthogonal_projection",
"set_like.coe_mem",
"submodule",
"submodule.coe_eq_zero",
"submodule.coe_norm",
"submodule.inner_rig... | The Pythagorean theorem, for an orthogonal projection. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id_eq_sum_orthogonal_projection_self_orthogonal_complement
[complete_space E] [complete_space K] :
continuous_linear_map.id 𝕜 E
= K.subtypeL.comp (orthogonal_projection K)
+ Kᗮ.subtypeL.comp (orthogonal_projection Kᗮ) | by { ext w, exact eq_sum_orthogonal_projection_self_orthogonal_complement K w } | lemma | id_eq_sum_orthogonal_projection_self_orthogonal_complement | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"complete_space",
"continuous_linear_map.id",
"eq_sum_orthogonal_projection_self_orthogonal_complement",
"orthogonal_projection"
] | In a complete space `E`, the projection maps onto a complete subspace `K` and its orthogonal
complement sum to the identity. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inner_orthogonal_projection_eq_of_mem_right [complete_space K] (u : K) (v : E) :
⟪orthogonal_projection K v, u⟫ = ⟪v, u⟫ | calc ⟪orthogonal_projection K v, u⟫
= ⟪(orthogonal_projection K v : E), u⟫ : K.coe_inner _ _
... = ⟪(orthogonal_projection K v : E), u⟫ + ⟪v - orthogonal_projection K v, u⟫ :
by rw [orthogonal_projection_inner_eq_zero _ _ (submodule.coe_mem _), add_zero]
... = ⟪v, u⟫ :
by rw [← inner_add_left, add_sub_c... | lemma | inner_orthogonal_projection_eq_of_mem_right | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"complete_space",
"inner_add_left",
"orthogonal_projection",
"orthogonal_projection_inner_eq_zero",
"submodule.coe_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_orthogonal_projection_eq_of_mem_left [complete_space K] (u : K) (v : E) :
⟪u, orthogonal_projection K v⟫ = ⟪(u : E), v⟫ | by rw [← inner_conj_symm, ← inner_conj_symm (u : E), inner_orthogonal_projection_eq_of_mem_right] | lemma | inner_orthogonal_projection_eq_of_mem_left | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"complete_space",
"inner_conj_symm",
"inner_orthogonal_projection_eq_of_mem_right",
"orthogonal_projection"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_orthogonal_projection_left_eq_right
[complete_space K] (u v : E) :
⟪↑(orthogonal_projection K u), v⟫ = ⟪u, orthogonal_projection K v⟫ | by rw [← inner_orthogonal_projection_eq_of_mem_left, inner_orthogonal_projection_eq_of_mem_right] | lemma | inner_orthogonal_projection_left_eq_right | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"complete_space",
"inner_orthogonal_projection_eq_of_mem_left",
"inner_orthogonal_projection_eq_of_mem_right",
"orthogonal_projection"
] | The orthogonal projection is self-adjoint. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthogonal_projection_is_symmetric
[complete_space K] :
(K.subtypeL ∘L orthogonal_projection K : E →ₗ[𝕜] E).is_symmetric | inner_orthogonal_projection_left_eq_right K | lemma | orthogonal_projection_is_symmetric | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"complete_space",
"inner_orthogonal_projection_left_eq_right",
"orthogonal_projection"
] | The orthogonal projection is symmetric. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
submodule.finrank_add_inf_finrank_orthogonal {K₁ K₂ : submodule 𝕜 E}
[finite_dimensional 𝕜 K₂] (h : K₁ ≤ K₂) :
finrank 𝕜 K₁ + finrank 𝕜 (K₁ᗮ ⊓ K₂ : submodule 𝕜 E) = finrank 𝕜 K₂ | begin
haveI := submodule.finite_dimensional_of_le h,
haveI := proper_is_R_or_C 𝕜 K₁,
have hd := submodule.finrank_sup_add_finrank_inf_eq K₁ (K₁ᗮ ⊓ K₂),
rw [←inf_assoc, (submodule.orthogonal_disjoint K₁).eq_bot, bot_inf_eq, finrank_bot,
submodule.sup_orthogonal_inf_of_complete_space h] at hd,
rw add_zer... | lemma | submodule.finrank_add_inf_finrank_orthogonal | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"bot_inf_eq",
"finite_dimensional",
"finrank_bot",
"submodule",
"submodule.finite_dimensional_of_le",
"submodule.finrank_sup_add_finrank_inf_eq",
"submodule.orthogonal_disjoint",
"submodule.sup_orthogonal_inf_of_complete_space"
] | Given a finite-dimensional subspace `K₂`, and a subspace `K₁`
containined in it, the dimensions of `K₁` and the intersection of its
orthogonal subspace with `K₂` add to that of `K₂`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
submodule.finrank_add_inf_finrank_orthogonal' {K₁ K₂ : submodule 𝕜 E}
[finite_dimensional 𝕜 K₂] (h : K₁ ≤ K₂) {n : ℕ} (h_dim : finrank 𝕜 K₁ + n = finrank 𝕜 K₂) :
finrank 𝕜 (K₁ᗮ ⊓ K₂ : submodule 𝕜 E) = n | by { rw ← add_right_inj (finrank 𝕜 K₁),
simp [submodule.finrank_add_inf_finrank_orthogonal h, h_dim] } | lemma | submodule.finrank_add_inf_finrank_orthogonal' | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"finite_dimensional",
"submodule",
"submodule.finrank_add_inf_finrank_orthogonal"
] | Given a finite-dimensional subspace `K₂`, and a subspace `K₁`
containined in it, the dimensions of `K₁` and the intersection of its
orthogonal subspace with `K₂` add to that of `K₂`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
submodule.finrank_add_finrank_orthogonal [finite_dimensional 𝕜 E] (K : submodule 𝕜 E) :
finrank 𝕜 K + finrank 𝕜 Kᗮ = finrank 𝕜 E | begin
convert submodule.finrank_add_inf_finrank_orthogonal (le_top : K ≤ ⊤) using 1,
{ rw inf_top_eq },
{ simp }
end | lemma | submodule.finrank_add_finrank_orthogonal | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"finite_dimensional",
"inf_top_eq",
"le_top",
"submodule",
"submodule.finrank_add_inf_finrank_orthogonal"
] | Given a finite-dimensional space `E` and subspace `K`, the dimensions of `K` and `Kᗮ` add to
that of `E`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
submodule.finrank_add_finrank_orthogonal' [finite_dimensional 𝕜 E] {K : submodule 𝕜 E} {n : ℕ}
(h_dim : finrank 𝕜 K + n = finrank 𝕜 E) :
finrank 𝕜 Kᗮ = n | by { rw ← add_right_inj (finrank 𝕜 K), simp [submodule.finrank_add_finrank_orthogonal, h_dim] } | lemma | submodule.finrank_add_finrank_orthogonal' | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"finite_dimensional",
"submodule",
"submodule.finrank_add_finrank_orthogonal"
] | Given a finite-dimensional space `E` and subspace `K`, the dimensions of `K` and `Kᗮ` add to
that of `E`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finrank_orthogonal_span_singleton {n : ℕ} [_i : fact (finrank 𝕜 E = n + 1)]
{v : E} (hv : v ≠ 0) :
finrank 𝕜 (𝕜 ∙ v)ᗮ = n | submodule.finrank_add_finrank_orthogonal' $ by simp [finrank_span_singleton hv, _i.elim, add_comm] | lemma | finrank_orthogonal_span_singleton | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"fact",
"finrank_span_singleton",
"submodule.finrank_add_finrank_orthogonal'"
] | In a finite-dimensional inner product space, the dimension of the orthogonal complement of the
span of a nonzero vector is one less than the dimension of the space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_isometry_equiv.reflections_generate_dim_aux [finite_dimensional ℝ F] {n : ℕ}
(φ : F ≃ₗᵢ[ℝ] F)
(hn : finrank ℝ (ker (continuous_linear_map.id ℝ F - φ))ᗮ ≤ n) :
∃ l : list F, l.length ≤ n ∧ φ = (l.map (λ v, reflection (ℝ ∙ v)ᗮ)).prod | begin
-- We prove this by strong induction on `n`, the dimension of the orthogonal complement of the
-- fixed subspace of the endomorphism `φ`
induction n with n IH generalizing φ,
{ -- Base case: `n = 0`, the fixed subspace is the whole space, so `φ = id`
refine ⟨[], rfl.le, show φ = 1, from _⟩,
have :... | lemma | linear_isometry_equiv.reflections_generate_dim_aux | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"continuous_linear_map.coe_sub",
"continuous_linear_map.id",
"continuous_linear_map.to_linear_map_eq_coe",
"exists_ne",
"finite_dimensional",
"finrank_eq_zero",
"le_zero_iff",
"linear_isometry_equiv.inner_map_map",
"linear_map.congr_fun",
"linear_map.sub_apply",
"linear_map.zero_apply",
"list.... | An element `φ` of the orthogonal group of `F` can be factored as a product of reflections, and
specifically at most as many reflections as the dimension of the complement of the fixed subspace
of `φ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_isometry_equiv.reflections_generate_dim [finite_dimensional ℝ F] (φ : F ≃ₗᵢ[ℝ] F) :
∃ l : list F, l.length ≤ finrank ℝ F ∧ φ = (l.map (λ v, reflection (ℝ ∙ v)ᗮ)).prod | let ⟨l, hl₁, hl₂⟩ := φ.reflections_generate_dim_aux le_rfl in
⟨l, hl₁.trans (submodule.finrank_le _), hl₂⟩ | lemma | linear_isometry_equiv.reflections_generate_dim | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"finite_dimensional",
"le_rfl",
"reflection",
"submodule.finrank_le"
] | The orthogonal group of `F` is generated by reflections; specifically each element `φ` of the
orthogonal group is a product of at most as many reflections as the dimension of `F`.
Special case of the **Cartan–Dieudonné theorem**. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_isometry_equiv.reflections_generate [finite_dimensional ℝ F] :
subgroup.closure (set.range (λ v : F, reflection (ℝ ∙ v)ᗮ)) = ⊤ | begin
rw subgroup.eq_top_iff',
intros φ,
rcases φ.reflections_generate_dim with ⟨l, _, rfl⟩,
apply (subgroup.closure _).list_prod_mem,
intros x hx,
rcases list.mem_map.mp hx with ⟨a, _, hax⟩,
exact subgroup.subset_closure ⟨a, hax⟩,
end | lemma | linear_isometry_equiv.reflections_generate | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"finite_dimensional",
"list_prod_mem",
"reflection",
"set.range",
"subgroup.closure",
"subgroup.eq_top_iff'",
"subgroup.subset_closure"
] | The orthogonal group of `F` is generated by reflections. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthogonal_family.is_internal_iff_of_is_complete [decidable_eq ι]
{V : ι → submodule 𝕜 E} (hV : orthogonal_family 𝕜 (λ i, V i) (λ i, (V i).subtypeₗᵢ))
(hc : is_complete (↑(supr V) : set E)) :
direct_sum.is_internal V ↔ (supr V)ᗮ = ⊥ | begin
haveI : complete_space ↥(supr V) := hc.complete_space_coe,
simp only [direct_sum.is_internal_submodule_iff_independent_and_supr_eq_top, hV.independent,
true_and, submodule.orthogonal_eq_bot_iff]
end | lemma | orthogonal_family.is_internal_iff_of_is_complete | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"complete_space",
"direct_sum.is_internal",
"direct_sum.is_internal_submodule_iff_independent_and_supr_eq_top",
"is_complete",
"orthogonal_family",
"submodule",
"submodule.orthogonal_eq_bot_iff",
"supr"
] | An orthogonal family of subspaces of `E` satisfies `direct_sum.is_internal` (that is,
they provide an internal direct sum decomposition of `E`) if and only if their span has trivial
orthogonal complement. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthogonal_family.is_internal_iff [decidable_eq ι] [finite_dimensional 𝕜 E]
{V : ι → submodule 𝕜 E} (hV : orthogonal_family 𝕜 (λ i, V i) (λ i, (V i).subtypeₗᵢ)) :
direct_sum.is_internal V ↔ (supr V)ᗮ = ⊥ | begin
haveI h := finite_dimensional.proper_is_R_or_C 𝕜 ↥(supr V),
exact hV.is_internal_iff_of_is_complete
(complete_space_coe_iff_is_complete.mp infer_instance)
end | lemma | orthogonal_family.is_internal_iff | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"direct_sum.is_internal",
"finite_dimensional",
"finite_dimensional.proper_is_R_or_C",
"orthogonal_family",
"submodule",
"supr"
] | An orthogonal family of subspaces of `E` satisfies `direct_sum.is_internal` (that is,
they provide an internal direct sum decomposition of `E`) if and only if their span has trivial
orthogonal complement. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthogonal_family.sum_projection_of_mem_supr [fintype ι]
{V : ι → submodule 𝕜 E} [∀ i, complete_space ↥(V i)]
(hV : orthogonal_family 𝕜 (λ i, V i) (λ i, (V i).subtypeₗᵢ)) (x : E) (hx : x ∈ supr V) :
∑ i, (orthogonal_projection (V i) x : E) = x | begin
refine submodule.supr_induction _ hx (λ i x hx, _) _ (λ x y hx hy, _),
{ refine (finset.sum_eq_single_of_mem i (finset.mem_univ _) $ λ j _ hij, _).trans
(orthogonal_projection_eq_self_iff.mpr hx),
rw [orthogonal_projection_mem_subspace_orthogonal_complement_eq_zero, submodule.coe_zero],
exact hV... | lemma | orthogonal_family.sum_projection_of_mem_supr | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"complete_space",
"congr_arg2",
"finset.mem_univ",
"fintype",
"orthogonal_family",
"orthogonal_projection",
"orthogonal_projection_mem_subspace_orthogonal_complement_eq_zero",
"submodule",
"submodule.coe_add",
"submodule.coe_zero",
"submodule.supr_induction",
"supr"
] | If `x` lies within an orthogonal family `v`, it can be expressed as a sum of projections. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthogonal_family.projection_direct_sum_coe_add_hom [decidable_eq ι]
{V : ι → submodule 𝕜 E} (hV : orthogonal_family 𝕜 (λ i, V i) (λ i, (V i).subtypeₗᵢ))
(x : ⨁ i, V i) (i : ι) [complete_space ↥(V i)] :
orthogonal_projection (V i) (direct_sum.coe_add_monoid_hom V x) = x i | begin
induction x using direct_sum.induction_on with j x x y hx hy,
{ simp },
{ simp_rw [direct_sum.coe_add_monoid_hom_of, direct_sum.of, dfinsupp.single_add_hom_apply],
obtain rfl | hij := decidable.eq_or_ne i j,
{ rw [orthogonal_projection_mem_subspace_eq_self, dfinsupp.single_eq_same] },
{ rw [orth... | lemma | orthogonal_family.projection_direct_sum_coe_add_hom | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"complete_space",
"congr_arg2",
"decidable.eq_or_ne",
"dfinsupp.add_apply",
"dfinsupp.single_eq_of_ne",
"dfinsupp.single_eq_same",
"direct_sum.coe_add_monoid_hom",
"direct_sum.coe_add_monoid_hom_of",
"direct_sum.induction_on",
"direct_sum.of",
"orthogonal_family",
"orthogonal_projection",
"o... | If a family of submodules is orthogonal, then the `orthogonal_projection` on a direct sum
is just the coefficient of that direct sum. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthogonal_family.decomposition [decidable_eq ι] [fintype ι] {V : ι → submodule 𝕜 E}
[∀ i, complete_space ↥(V i)]
(hV : orthogonal_family 𝕜 (λ i, V i) (λ i, (V i).subtypeₗᵢ)) (h : supr V = ⊤) :
direct_sum.decomposition V | { decompose' := λ x, dfinsupp.equiv_fun_on_fintype.symm $ λ i, orthogonal_projection (V i) x,
left_inv := λ x, begin
dsimp only,
letI := λ i, classical.dec_eq (V i),
rw [direct_sum.coe_add_monoid_hom, direct_sum.to_add_monoid, dfinsupp.lift_add_hom_apply,
dfinsupp.sum_add_hom_apply, dfinsupp.sum_eq_... | def | orthogonal_family.decomposition | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"classical.dec_eq",
"complete_space",
"dfinsupp.equiv_fun_on_fintype_symm_coe",
"dfinsupp.sum_add_hom_apply",
"direct_sum.coe_add_monoid_hom",
"direct_sum.decomposition",
"direct_sum.to_add_monoid",
"equiv.apply_symm_apply",
"fintype",
"orthogonal_family",
"orthogonal_projection",
"submodule",... | If a family of submodules is orthogonal and they span the whole space, then the orthogonal
projection provides a means to decompose the space into its submodules.
The projection function is `decompose V x i = orthogonal_projection (V i) x`.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
maximal_orthonormal_iff_orthogonal_complement_eq_bot (hv : orthonormal 𝕜 (coe : v → E)) :
(∀ u ⊇ v, orthonormal 𝕜 (coe : u → E) → u = v) ↔ (span 𝕜 v)ᗮ = ⊥ | begin
rw submodule.eq_bot_iff,
split,
{ contrapose!,
-- ** direction 1: nonempty orthogonal complement implies nonmaximal
rintros ⟨x, hx', hx⟩,
-- take a nonzero vector and normalize it
let e := (‖x‖⁻¹ : 𝕜) • x,
have he : ‖e‖ = 1 := by simp [e, norm_smul_inv_norm hx],
have he' : e ∈ (span... | lemma | maximal_orthonormal_iff_orthogonal_complement_eq_bot | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"finsupp.mem_span_image_iff_total",
"finsupp.supported",
"finsupp.total",
"inner_eq_zero_symm",
"norm_smul_inv_norm",
"orthonormal",
"submodule.eq_bot_iff",
"submodule.subset_span"
] | An orthonormal set in an `inner_product_space` is maximal, if and only if the orthogonal
complement of its span is empty. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
maximal_orthonormal_iff_basis_of_finite_dimensional
(hv : orthonormal 𝕜 (coe : v → E)) :
(∀ u ⊇ v, orthonormal 𝕜 (coe : u → E) → u = v) ↔ ∃ b : basis v 𝕜 E, ⇑b = coe | begin
haveI := proper_is_R_or_C 𝕜 (span 𝕜 v),
rw maximal_orthonormal_iff_orthogonal_complement_eq_bot hv,
have hv_compl : is_complete (span 𝕜 v : set E) := (span 𝕜 v).complete_of_finite_dimensional,
rw submodule.orthogonal_eq_bot_iff,
have hv_coe : range (coe : v → E) = v := by simp,
split,
{ refine λ... | lemma | maximal_orthonormal_iff_basis_of_finite_dimensional | analysis.inner_product_space | src/analysis/inner_product_space/projection.lean | [
"algebra.direct_sum.decomposition",
"analysis.convex.basic",
"analysis.inner_product_space.orthogonal",
"analysis.inner_product_space.symmetric",
"analysis.normed_space.is_R_or_C",
"data.is_R_or_C.lemmas"
] | [
"basis",
"basis.coe_mk",
"is_complete",
"maximal_orthonormal_iff_orthogonal_complement_eq_bot",
"orthonormal",
"submodule.orthogonal_eq_bot_iff"
] | An orthonormal set in a finite-dimensional `inner_product_space` is maximal, if and only if it
is a basis. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rayleigh_smul (x : E) {c : 𝕜} (hc : c ≠ 0) :
rayleigh_quotient (c • x) = rayleigh_quotient x | begin
by_cases hx : x = 0,
{ simp [hx] },
have : ‖c‖ ≠ 0 := by simp [hc],
have : ‖x‖ ≠ 0 := by simp [hx],
field_simp [norm_smul, T.re_apply_inner_self_smul],
ring
end | lemma | continuous_linear_map.rayleigh_smul | analysis.inner_product_space | src/analysis/inner_product_space/rayleigh.lean | [
"analysis.inner_product_space.calculus",
"analysis.inner_product_space.dual",
"analysis.inner_product_space.adjoint",
"analysis.calculus.lagrange_multipliers",
"linear_algebra.eigenspace.basic"
] | [
"norm_smul",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_rayleigh_eq_image_rayleigh_sphere {r : ℝ} (hr : 0 < r) :
rayleigh_quotient '' {0}ᶜ = rayleigh_quotient '' (sphere 0 r) | begin
ext a,
split,
{ rintros ⟨x, (hx : x ≠ 0), hxT⟩,
have : ‖x‖ ≠ 0 := by simp [hx],
let c : 𝕜 := ↑‖x‖⁻¹ * r,
have : c ≠ 0 := by simp [c, hx, hr.ne'],
refine ⟨c • x, _, _⟩,
{ field_simp [norm_smul, abs_of_pos hr] },
{ rw T.rayleigh_smul x this,
exact hxT } },
{ rintros ⟨x, hx, hx... | lemma | continuous_linear_map.image_rayleigh_eq_image_rayleigh_sphere | analysis.inner_product_space | src/analysis/inner_product_space/rayleigh.lean | [
"analysis.inner_product_space.calculus",
"analysis.inner_product_space.dual",
"analysis.inner_product_space.adjoint",
"analysis.calculus.lagrange_multipliers",
"linear_algebra.eigenspace.basic"
] | [
"abs_of_pos",
"norm_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
supr_rayleigh_eq_supr_rayleigh_sphere {r : ℝ} (hr : 0 < r) :
(⨆ x : {x : E // x ≠ 0}, rayleigh_quotient x) = ⨆ x : sphere (0:E) r, rayleigh_quotient x | show (⨆ x : ({0} : set E)ᶜ, rayleigh_quotient x) = _,
by simp only [←@Sup_image' _ _ _ _ rayleigh_quotient, T.image_rayleigh_eq_image_rayleigh_sphere hr] | lemma | continuous_linear_map.supr_rayleigh_eq_supr_rayleigh_sphere | analysis.inner_product_space | src/analysis/inner_product_space/rayleigh.lean | [
"analysis.inner_product_space.calculus",
"analysis.inner_product_space.dual",
"analysis.inner_product_space.adjoint",
"analysis.calculus.lagrange_multipliers",
"linear_algebra.eigenspace.basic"
] | [
"Sup_image'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
infi_rayleigh_eq_infi_rayleigh_sphere {r : ℝ} (hr : 0 < r) :
(⨅ x : {x : E // x ≠ 0}, rayleigh_quotient x) = ⨅ x : sphere (0:E) r, rayleigh_quotient x | show (⨅ x : ({0} : set E)ᶜ, rayleigh_quotient x) = _,
by simp only [←@Inf_image' _ _ _ _ rayleigh_quotient, T.image_rayleigh_eq_image_rayleigh_sphere hr] | lemma | continuous_linear_map.infi_rayleigh_eq_infi_rayleigh_sphere | analysis.inner_product_space | src/analysis/inner_product_space/rayleigh.lean | [
"analysis.inner_product_space.calculus",
"analysis.inner_product_space.dual",
"analysis.inner_product_space.adjoint",
"analysis.calculus.lagrange_multipliers",
"linear_algebra.eigenspace.basic"
] | [
"Inf_image'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.linear_map.is_symmetric.has_strict_fderiv_at_re_apply_inner_self
{T : F →L[ℝ] F} (hT : (T : F →ₗ[ℝ] F).is_symmetric) (x₀ : F) :
has_strict_fderiv_at T.re_apply_inner_self (_root_.bit0 (innerSL ℝ (T x₀))) x₀ | begin
convert T.has_strict_fderiv_at.inner _ (has_strict_fderiv_at_id x₀),
ext y,
simp_rw [_root_.bit0, continuous_linear_map.comp_apply, continuous_linear_map.add_apply,
innerSL_apply, fderiv_inner_clm_apply, id.def, continuous_linear_map.prod_apply,
continuous_linear_map.id_apply, hT.apply_clm x₀ y, rea... | lemma | linear_map.is_symmetric.has_strict_fderiv_at_re_apply_inner_self | analysis.inner_product_space | src/analysis/inner_product_space/rayleigh.lean | [
"analysis.inner_product_space.calculus",
"analysis.inner_product_space.dual",
"analysis.inner_product_space.adjoint",
"analysis.calculus.lagrange_multipliers",
"linear_algebra.eigenspace.basic"
] | [
"continuous_linear_map.add_apply",
"continuous_linear_map.comp_apply",
"continuous_linear_map.id_apply",
"continuous_linear_map.prod_apply",
"fderiv_inner_clm_apply",
"has_strict_fderiv_at",
"has_strict_fderiv_at_id",
"innerSL",
"innerSL_apply",
"real_inner_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
linearly_dependent_of_is_local_extr_on (hT : is_self_adjoint T)
{x₀ : F} (hextr : is_local_extr_on T.re_apply_inner_self (sphere (0:F) ‖x₀‖) x₀) :
∃ a b : ℝ, (a, b) ≠ 0 ∧ a • x₀ + b • T x₀ = 0 | begin
have H : is_local_extr_on T.re_apply_inner_self {x : F | ‖x‖ ^ 2 = ‖x₀‖ ^ 2} x₀,
{ convert hextr,
ext x,
simp [dist_eq_norm] },
-- find Lagrange multipliers for the function `T.re_apply_inner_self` and the
-- hypersurface-defining function `λ x, ‖x‖ ^ 2`
obtain ⟨a, b, h₁, h₂⟩ := is_local_extr_on... | lemma | is_self_adjoint.linearly_dependent_of_is_local_extr_on | analysis.inner_product_space | src/analysis/inner_product_space/rayleigh.lean | [
"analysis.inner_product_space.calculus",
"analysis.inner_product_space.dual",
"analysis.inner_product_space.adjoint",
"analysis.calculus.lagrange_multipliers",
"linear_algebra.eigenspace.basic"
] | [
"add_smul",
"has_strict_fderiv_at_norm_sq",
"innerSL",
"inner_product_space.to_dual_map",
"is_local_extr_on",
"is_local_extr_on.exists_multipliers_of_has_strict_fderiv_at_1d",
"is_self_adjoint",
"linear_isometry.map_add",
"linear_isometry.map_smul",
"linear_isometry.map_zero",
"one_smul",
"smu... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_smul_self_of_is_local_extr_on_real (hT : is_self_adjoint T)
{x₀ : F} (hextr : is_local_extr_on T.re_apply_inner_self (sphere (0:F) ‖x₀‖) x₀) :
T x₀ = (rayleigh_quotient x₀) • x₀ | begin
obtain ⟨a, b, h₁, h₂⟩ := hT.linearly_dependent_of_is_local_extr_on hextr,
by_cases hx₀ : x₀ = 0,
{ simp [hx₀] },
by_cases hb : b = 0,
{ have : a ≠ 0 := by simpa [hb] using h₁,
refine absurd _ hx₀,
apply smul_right_injective F this,
simpa [hb] using h₂ },
let c : ℝ := - b⁻¹ * a,
have hc :... | lemma | is_self_adjoint.eq_smul_self_of_is_local_extr_on_real | analysis.inner_product_space | src/analysis/inner_product_space/rayleigh.lean | [
"analysis.inner_product_space.calculus",
"analysis.inner_product_space.dual",
"analysis.inner_product_space.adjoint",
"analysis.calculus.lagrange_multipliers",
"linear_algebra.eigenspace.basic"
] | [
"inner_smul_left",
"is_local_extr_on",
"is_self_adjoint",
"mul_comm",
"real_inner_self_eq_norm_mul_norm",
"smul_right_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_smul_self_of_is_local_extr_on (hT : is_self_adjoint T) {x₀ : E}
(hextr : is_local_extr_on T.re_apply_inner_self (sphere (0:E) ‖x₀‖) x₀) :
T x₀ = (↑(rayleigh_quotient x₀) : 𝕜) • x₀ | begin
letI := inner_product_space.is_R_or_C_to_real 𝕜 E,
let hSA := hT.is_symmetric.restrict_scalars.to_self_adjoint.prop,
exact hSA.eq_smul_self_of_is_local_extr_on_real hextr,
end | lemma | is_self_adjoint.eq_smul_self_of_is_local_extr_on | analysis.inner_product_space | src/analysis/inner_product_space/rayleigh.lean | [
"analysis.inner_product_space.calculus",
"analysis.inner_product_space.dual",
"analysis.inner_product_space.adjoint",
"analysis.calculus.lagrange_multipliers",
"linear_algebra.eigenspace.basic"
] | [
"inner_product_space.is_R_or_C_to_real",
"is_local_extr_on",
"is_self_adjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_eigenvector_of_is_local_extr_on (hT : is_self_adjoint T) {x₀ : E}
(hx₀ : x₀ ≠ 0) (hextr : is_local_extr_on T.re_apply_inner_self (sphere (0:E) ‖x₀‖) x₀) :
has_eigenvector (T : E →ₗ[𝕜] E) ↑(rayleigh_quotient x₀) x₀ | begin
refine ⟨_, hx₀⟩,
rw module.End.mem_eigenspace_iff,
exact hT.eq_smul_self_of_is_local_extr_on hextr
end | lemma | is_self_adjoint.has_eigenvector_of_is_local_extr_on | analysis.inner_product_space | src/analysis/inner_product_space/rayleigh.lean | [
"analysis.inner_product_space.calculus",
"analysis.inner_product_space.dual",
"analysis.inner_product_space.adjoint",
"analysis.calculus.lagrange_multipliers",
"linear_algebra.eigenspace.basic"
] | [
"is_local_extr_on",
"is_self_adjoint",
"module.End.mem_eigenspace_iff"
] | For a self-adjoint operator `T`, a local extremum of the Rayleigh quotient of `T` on a sphere
centred at the origin is an eigenvector of `T`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_eigenvector_of_is_max_on (hT : is_self_adjoint T) {x₀ : E}
(hx₀ : x₀ ≠ 0) (hextr : is_max_on T.re_apply_inner_self (sphere (0:E) ‖x₀‖) x₀) :
has_eigenvector (T : E →ₗ[𝕜] E) ↑(⨆ x : {x : E // x ≠ 0}, rayleigh_quotient x) x₀ | begin
convert hT.has_eigenvector_of_is_local_extr_on hx₀ (or.inr hextr.localize),
have hx₀' : 0 < ‖x₀‖ := by simp [hx₀],
have hx₀'' : x₀ ∈ sphere (0:E) (‖x₀‖) := by simp,
rw T.supr_rayleigh_eq_supr_rayleigh_sphere hx₀',
refine is_max_on.supr_eq hx₀'' _,
intros x hx,
dsimp,
have : ‖x‖ = ‖x₀‖ := by simpa ... | lemma | is_self_adjoint.has_eigenvector_of_is_max_on | analysis.inner_product_space | src/analysis/inner_product_space/rayleigh.lean | [
"analysis.inner_product_space.calculus",
"analysis.inner_product_space.dual",
"analysis.inner_product_space.adjoint",
"analysis.calculus.lagrange_multipliers",
"linear_algebra.eigenspace.basic"
] | [
"div_le_div_of_le",
"is_max_on",
"is_max_on.supr_eq",
"is_self_adjoint",
"sq_nonneg"
] | For a self-adjoint operator `T`, a maximum of the Rayleigh quotient of `T` on a sphere centred
at the origin is an eigenvector of `T`, with eigenvalue the global supremum of the Rayleigh
quotient. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_eigenvector_of_is_min_on (hT : is_self_adjoint T) {x₀ : E}
(hx₀ : x₀ ≠ 0) (hextr : is_min_on T.re_apply_inner_self (sphere (0:E) ‖x₀‖) x₀) :
has_eigenvector (T : E →ₗ[𝕜] E) ↑(⨅ x : {x : E // x ≠ 0}, rayleigh_quotient x) x₀ | begin
convert hT.has_eigenvector_of_is_local_extr_on hx₀ (or.inl hextr.localize),
have hx₀' : 0 < ‖x₀‖ := by simp [hx₀],
have hx₀'' : x₀ ∈ sphere (0:E) (‖x₀‖) := by simp,
rw T.infi_rayleigh_eq_infi_rayleigh_sphere hx₀',
refine is_min_on.infi_eq hx₀'' _,
intros x hx,
dsimp,
have : ‖x‖ = ‖x₀‖ := by simpa ... | lemma | is_self_adjoint.has_eigenvector_of_is_min_on | analysis.inner_product_space | src/analysis/inner_product_space/rayleigh.lean | [
"analysis.inner_product_space.calculus",
"analysis.inner_product_space.dual",
"analysis.inner_product_space.adjoint",
"analysis.calculus.lagrange_multipliers",
"linear_algebra.eigenspace.basic"
] | [
"div_le_div_of_le",
"is_min_on",
"is_min_on.infi_eq",
"is_self_adjoint",
"sq_nonneg"
] | For a self-adjoint operator `T`, a minimum of the Rayleigh quotient of `T` on a sphere centred
at the origin is an eigenvector of `T`, with eigenvalue the global infimum of the Rayleigh
quotient. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_eigenvalue_supr_of_finite_dimensional (hT : T.is_symmetric) :
has_eigenvalue T ↑(⨆ x : {x : E // x ≠ 0}, is_R_or_C.re ⟪T x, x⟫ / ‖(x:E)‖ ^ 2) | begin
haveI := finite_dimensional.proper_is_R_or_C 𝕜 E,
let T' := hT.to_self_adjoint,
obtain ⟨x, hx⟩ : ∃ x : E, x ≠ 0 := exists_ne 0,
have H₁ : is_compact (sphere (0:E) ‖x‖) := is_compact_sphere _ _,
have H₂ : (sphere (0:E) ‖x‖).nonempty := ⟨x, by simp⟩,
-- key point: in finite dimension, a continuous func... | lemma | linear_map.is_symmetric.has_eigenvalue_supr_of_finite_dimensional | analysis.inner_product_space | src/analysis/inner_product_space/rayleigh.lean | [
"analysis.inner_product_space.calculus",
"analysis.inner_product_space.dual",
"analysis.inner_product_space.adjoint",
"analysis.calculus.lagrange_multipliers",
"linear_algebra.eigenspace.basic"
] | [
"exists_ne",
"finite_dimensional.proper_is_R_or_C",
"is_compact",
"is_compact_sphere",
"is_max_on",
"norm_eq_zero"
] | The supremum of the Rayleigh quotient of a symmetric operator `T` on a nontrivial
finite-dimensional vector space is an eigenvalue for that operator. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_eigenvalue_infi_of_finite_dimensional (hT : T.is_symmetric) :
has_eigenvalue T ↑(⨅ x : {x : E // x ≠ 0}, is_R_or_C.re ⟪T x, x⟫ / ‖(x:E)‖ ^ 2) | begin
haveI := finite_dimensional.proper_is_R_or_C 𝕜 E,
let T' := hT.to_self_adjoint,
obtain ⟨x, hx⟩ : ∃ x : E, x ≠ 0 := exists_ne 0,
have H₁ : is_compact (sphere (0:E) ‖x‖) := is_compact_sphere _ _,
have H₂ : (sphere (0:E) ‖x‖).nonempty := ⟨x, by simp⟩,
-- key point: in finite dimension, a continuous func... | lemma | linear_map.is_symmetric.has_eigenvalue_infi_of_finite_dimensional | analysis.inner_product_space | src/analysis/inner_product_space/rayleigh.lean | [
"analysis.inner_product_space.calculus",
"analysis.inner_product_space.dual",
"analysis.inner_product_space.adjoint",
"analysis.calculus.lagrange_multipliers",
"linear_algebra.eigenspace.basic"
] | [
"exists_ne",
"finite_dimensional.proper_is_R_or_C",
"is_compact",
"is_compact_sphere",
"is_min_on",
"norm_eq_zero"
] | The infimum of the Rayleigh quotient of a symmetric operator `T` on a nontrivial
finite-dimensional vector space is an eigenvalue for that operator. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subsingleton_of_no_eigenvalue_finite_dimensional
(hT : T.is_symmetric) (hT' : ∀ μ : 𝕜, module.End.eigenspace (T : E →ₗ[𝕜] E) μ = ⊥) :
subsingleton E | (subsingleton_or_nontrivial E).resolve_right
(λ h, by exactI absurd (hT' _) hT.has_eigenvalue_supr_of_finite_dimensional) | lemma | linear_map.is_symmetric.subsingleton_of_no_eigenvalue_finite_dimensional | analysis.inner_product_space | src/analysis/inner_product_space/rayleigh.lean | [
"analysis.inner_product_space.calculus",
"analysis.inner_product_space.dual",
"analysis.inner_product_space.adjoint",
"analysis.calculus.lagrange_multipliers",
"linear_algebra.eigenspace.basic"
] | [
"module.End.eigenspace",
"subsingleton_or_nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
invariant_orthogonal_eigenspace (μ : 𝕜) (v : E) (hv : v ∈ (eigenspace T μ)ᗮ) :
T v ∈ (eigenspace T μ)ᗮ | begin
intros w hw,
have : T w = (μ:𝕜) • w := by rwa mem_eigenspace_iff at hw,
simp [← hT w, this, inner_smul_left, hv w hw]
end | lemma | linear_map.is_symmetric.invariant_orthogonal_eigenspace | analysis.inner_product_space | src/analysis/inner_product_space/spectrum.lean | [
"analysis.inner_product_space.rayleigh",
"analysis.inner_product_space.pi_L2",
"algebra.direct_sum.decomposition",
"linear_algebra.eigenspace.minpoly"
] | [
"inner_smul_left"
] | A self-adjoint operator preserves orthogonal complements of its eigenspaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
conj_eigenvalue_eq_self {μ : 𝕜} (hμ : has_eigenvalue T μ) : conj μ = μ | begin
obtain ⟨v, hv₁, hv₂⟩ := hμ.exists_has_eigenvector,
rw mem_eigenspace_iff at hv₁,
simpa [hv₂, inner_smul_left, inner_smul_right, hv₁] using hT v v
end | lemma | linear_map.is_symmetric.conj_eigenvalue_eq_self | analysis.inner_product_space | src/analysis/inner_product_space/spectrum.lean | [
"analysis.inner_product_space.rayleigh",
"analysis.inner_product_space.pi_L2",
"algebra.direct_sum.decomposition",
"linear_algebra.eigenspace.minpoly"
] | [
"inner_smul_left",
"inner_smul_right"
] | The eigenvalues of a self-adjoint operator are real. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthogonal_family_eigenspaces :
orthogonal_family 𝕜 (λ μ, eigenspace T μ) (λ μ, (eigenspace T μ).subtypeₗᵢ) | begin
rintros μ ν hμν ⟨v, hv⟩ ⟨w, hw⟩,
by_cases hv' : v = 0,
{ simp [hv'] },
have H := hT.conj_eigenvalue_eq_self (has_eigenvalue_of_has_eigenvector ⟨hv, hv'⟩),
rw mem_eigenspace_iff at hv hw,
refine or.resolve_left _ hμν.symm,
simpa [inner_smul_left, inner_smul_right, hv, hw, H] using (hT v w).symm
end | lemma | linear_map.is_symmetric.orthogonal_family_eigenspaces | analysis.inner_product_space | src/analysis/inner_product_space/spectrum.lean | [
"analysis.inner_product_space.rayleigh",
"analysis.inner_product_space.pi_L2",
"algebra.direct_sum.decomposition",
"linear_algebra.eigenspace.minpoly"
] | [
"inner_smul_left",
"inner_smul_right",
"orthogonal_family"
] | The eigenspaces of a self-adjoint operator are mutually orthogonal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthogonal_family_eigenspaces' :
orthogonal_family 𝕜 (λ μ : eigenvalues T, eigenspace T μ) (λ μ, (eigenspace T μ).subtypeₗᵢ) | hT.orthogonal_family_eigenspaces.comp subtype.coe_injective | lemma | linear_map.is_symmetric.orthogonal_family_eigenspaces' | analysis.inner_product_space | src/analysis/inner_product_space/spectrum.lean | [
"analysis.inner_product_space.rayleigh",
"analysis.inner_product_space.pi_L2",
"algebra.direct_sum.decomposition",
"linear_algebra.eigenspace.minpoly"
] | [
"orthogonal_family",
"subtype.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
orthogonal_supr_eigenspaces_invariant ⦃v : E⦄ (hv : v ∈ (⨆ μ, eigenspace T μ)ᗮ) :
T v ∈ (⨆ μ, eigenspace T μ)ᗮ | begin
rw ← submodule.infi_orthogonal at ⊢ hv,
exact T.infi_invariant hT.invariant_orthogonal_eigenspace v hv
end | lemma | linear_map.is_symmetric.orthogonal_supr_eigenspaces_invariant | analysis.inner_product_space | src/analysis/inner_product_space/spectrum.lean | [
"analysis.inner_product_space.rayleigh",
"analysis.inner_product_space.pi_L2",
"algebra.direct_sum.decomposition",
"linear_algebra.eigenspace.minpoly"
] | [
"submodule.infi_orthogonal"
] | The mutual orthogonal complement of the eigenspaces of a self-adjoint operator on an inner
product space is an invariant subspace of the operator. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthogonal_supr_eigenspaces (μ : 𝕜) :
eigenspace (T.restrict hT.orthogonal_supr_eigenspaces_invariant) μ = ⊥ | begin
set p : submodule 𝕜 E := (⨆ μ, eigenspace T μ)ᗮ,
refine eigenspace_restrict_eq_bot hT.orthogonal_supr_eigenspaces_invariant _,
have H₂ : eigenspace T μ ⟂ p := (submodule.is_ortho_orthogonal_right _).mono_left (le_supr _ _),
exact H₂.disjoint
end | lemma | linear_map.is_symmetric.orthogonal_supr_eigenspaces | analysis.inner_product_space | src/analysis/inner_product_space/spectrum.lean | [
"analysis.inner_product_space.rayleigh",
"analysis.inner_product_space.pi_L2",
"algebra.direct_sum.decomposition",
"linear_algebra.eigenspace.minpoly"
] | [
"le_supr",
"submodule",
"submodule.is_ortho_orthogonal_right"
] | The mutual orthogonal complement of the eigenspaces of a self-adjoint operator on an inner
product space has no eigenvalues. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthogonal_supr_eigenspaces_eq_bot : (⨆ μ, eigenspace T μ)ᗮ = ⊥ | begin
have hT' : is_symmetric _ := hT.restrict_invariant hT.orthogonal_supr_eigenspaces_invariant,
-- a self-adjoint operator on a nontrivial inner product space has an eigenvalue
haveI := hT'.subsingleton_of_no_eigenvalue_finite_dimensional hT.orthogonal_supr_eigenspaces,
exact submodule.eq_bot_of_subsingleton... | lemma | linear_map.is_symmetric.orthogonal_supr_eigenspaces_eq_bot | analysis.inner_product_space | src/analysis/inner_product_space/spectrum.lean | [
"analysis.inner_product_space.rayleigh",
"analysis.inner_product_space.pi_L2",
"algebra.direct_sum.decomposition",
"linear_algebra.eigenspace.minpoly"
] | [
"submodule.eq_bot_of_subsingleton"
] | The mutual orthogonal complement of the eigenspaces of a self-adjoint operator on a
finite-dimensional inner product space is trivial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
orthogonal_supr_eigenspaces_eq_bot' : (⨆ μ : eigenvalues T, eigenspace T μ)ᗮ = ⊥ | show (⨆ μ : {μ // (eigenspace T μ) ≠ ⊥}, eigenspace T μ)ᗮ = ⊥,
by rw [supr_ne_bot_subtype, hT.orthogonal_supr_eigenspaces_eq_bot] | lemma | linear_map.is_symmetric.orthogonal_supr_eigenspaces_eq_bot' | analysis.inner_product_space | src/analysis/inner_product_space/spectrum.lean | [
"analysis.inner_product_space.rayleigh",
"analysis.inner_product_space.pi_L2",
"algebra.direct_sum.decomposition",
"linear_algebra.eigenspace.minpoly"
] | [
"supr_ne_bot_subtype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
direct_sum_decomposition [hT : fact T.is_symmetric] :
direct_sum.decomposition (λ μ : eigenvalues T, eigenspace T μ) | begin
haveI h : ∀ μ : eigenvalues T, complete_space (eigenspace T μ) := λ μ, by apply_instance,
exact hT.out.orthogonal_family_eigenspaces'.decomposition
(submodule.orthogonal_eq_bot_iff.mp hT.out.orthogonal_supr_eigenspaces_eq_bot'),
end | instance | linear_map.is_symmetric.direct_sum_decomposition | analysis.inner_product_space | src/analysis/inner_product_space/spectrum.lean | [
"analysis.inner_product_space.rayleigh",
"analysis.inner_product_space.pi_L2",
"algebra.direct_sum.decomposition",
"linear_algebra.eigenspace.minpoly"
] | [
"complete_space",
"direct_sum.decomposition",
"fact"
] | The eigenspaces of a self-adjoint operator on a finite-dimensional inner product space `E` gives
an internal direct sum decomposition of `E`.
Note this takes `hT` as a `fact` to allow it to be an instance. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
direct_sum_decompose_apply [hT : fact T.is_symmetric] (x : E) (μ : eigenvalues T) :
direct_sum.decompose (λ μ : eigenvalues T, eigenspace T μ) x μ
= orthogonal_projection (eigenspace T μ) x | rfl | lemma | linear_map.is_symmetric.direct_sum_decompose_apply | analysis.inner_product_space | src/analysis/inner_product_space/spectrum.lean | [
"analysis.inner_product_space.rayleigh",
"analysis.inner_product_space.pi_L2",
"algebra.direct_sum.decomposition",
"linear_algebra.eigenspace.minpoly"
] | [
"direct_sum.decompose",
"fact",
"orthogonal_projection"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
direct_sum_is_internal :
direct_sum.is_internal (λ μ : eigenvalues T, eigenspace T μ) | hT.orthogonal_family_eigenspaces'.is_internal_iff.mpr
hT.orthogonal_supr_eigenspaces_eq_bot' | lemma | linear_map.is_symmetric.direct_sum_is_internal | analysis.inner_product_space | src/analysis/inner_product_space/spectrum.lean | [
"analysis.inner_product_space.rayleigh",
"analysis.inner_product_space.pi_L2",
"algebra.direct_sum.decomposition",
"linear_algebra.eigenspace.minpoly"
] | [
"direct_sum.is_internal"
] | The eigenspaces of a self-adjoint operator on a finite-dimensional inner product space `E` gives
an internal direct sum decomposition of `E`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diagonalization : E ≃ₗᵢ[𝕜] pi_Lp 2 (λ μ : eigenvalues T, eigenspace T μ) | hT.direct_sum_is_internal.isometry_L2_of_orthogonal_family
hT.orthogonal_family_eigenspaces' | def | linear_map.is_symmetric.diagonalization | analysis.inner_product_space | src/analysis/inner_product_space/spectrum.lean | [
"analysis.inner_product_space.rayleigh",
"analysis.inner_product_space.pi_L2",
"algebra.direct_sum.decomposition",
"linear_algebra.eigenspace.minpoly"
] | [
"pi_Lp"
] | Isometry from an inner product space `E` to the direct sum of the eigenspaces of some
self-adjoint operator `T` on `E`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diagonalization_symm_apply (w : pi_Lp 2 (λ μ : eigenvalues T, eigenspace T μ)) :
hT.diagonalization.symm w = ∑ μ, w μ | hT.direct_sum_is_internal.isometry_L2_of_orthogonal_family_symm_apply
hT.orthogonal_family_eigenspaces' w | lemma | linear_map.is_symmetric.diagonalization_symm_apply | analysis.inner_product_space | src/analysis/inner_product_space/spectrum.lean | [
"analysis.inner_product_space.rayleigh",
"analysis.inner_product_space.pi_L2",
"algebra.direct_sum.decomposition",
"linear_algebra.eigenspace.minpoly"
] | [
"pi_Lp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diagonalization_apply_self_apply (v : E) (μ : eigenvalues T) :
hT.diagonalization (T v) μ = (μ : 𝕜) • hT.diagonalization v μ | begin
suffices : ∀ w : pi_Lp 2 (λ μ : eigenvalues T, eigenspace T μ),
(T (hT.diagonalization.symm w)) = hT.diagonalization.symm (λ μ, (μ : 𝕜) • w μ),
{ simpa only [linear_isometry_equiv.symm_apply_apply, linear_isometry_equiv.apply_symm_apply]
using congr_arg (λ w, hT.diagonalization w μ) (this (hT.diago... | lemma | linear_map.is_symmetric.diagonalization_apply_self_apply | analysis.inner_product_space | src/analysis/inner_product_space/spectrum.lean | [
"analysis.inner_product_space.rayleigh",
"analysis.inner_product_space.pi_L2",
"algebra.direct_sum.decomposition",
"linear_algebra.eigenspace.minpoly"
] | [
"linear_isometry_equiv.apply_symm_apply",
"linear_isometry_equiv.symm_apply_apply",
"pi_Lp",
"submodule.coe_smul_of_tower"
] | *Diagonalization theorem*, *spectral theorem*; version 1: A self-adjoint operator `T` on a
finite-dimensional inner product space `E` acts diagonally on the decomposition of `E` into the
direct sum of the eigenspaces of `T`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eigenvector_basis : orthonormal_basis (fin n) 𝕜 E | hT.direct_sum_is_internal.subordinate_orthonormal_basis hn
hT.orthogonal_family_eigenspaces' | def | linear_map.is_symmetric.eigenvector_basis | analysis.inner_product_space | src/analysis/inner_product_space/spectrum.lean | [
"analysis.inner_product_space.rayleigh",
"analysis.inner_product_space.pi_L2",
"algebra.direct_sum.decomposition",
"linear_algebra.eigenspace.minpoly"
] | [
"orthonormal_basis"
] | A choice of orthonormal basis of eigenvectors for self-adjoint operator `T` on a
finite-dimensional inner product space `E`.
TODO Postcompose with a permutation so that these eigenvectors are listed in increasing order of
eigenvalue. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eigenvalues (i : fin n) : ℝ | @is_R_or_C.re 𝕜 _ $
hT.direct_sum_is_internal.subordinate_orthonormal_basis_index hn i
hT.orthogonal_family_eigenspaces' | def | linear_map.is_symmetric.eigenvalues | analysis.inner_product_space | src/analysis/inner_product_space/spectrum.lean | [
"analysis.inner_product_space.rayleigh",
"analysis.inner_product_space.pi_L2",
"algebra.direct_sum.decomposition",
"linear_algebra.eigenspace.minpoly"
] | [] | The sequence of real eigenvalues associated to the standard orthonormal basis of eigenvectors
for a self-adjoint operator `T` on `E`.
TODO Postcompose with a permutation so that these eigenvalues are listed in increasing order. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_eigenvector_eigenvector_basis (i : fin n) :
has_eigenvector T (hT.eigenvalues hn i) (hT.eigenvector_basis hn i) | begin
let v : E := hT.eigenvector_basis hn i,
let μ : 𝕜 := hT.direct_sum_is_internal.subordinate_orthonormal_basis_index
hn i hT.orthogonal_family_eigenspaces',
simp_rw [eigenvalues],
change has_eigenvector T (is_R_or_C.re μ) v,
have key : has_eigenvector T μ v,
{ have H₁ : v ∈ eigenspace T μ,
{ si... | lemma | linear_map.is_symmetric.has_eigenvector_eigenvector_basis | analysis.inner_product_space | src/analysis/inner_product_space/spectrum.lean | [
"analysis.inner_product_space.rayleigh",
"analysis.inner_product_space.pi_L2",
"algebra.direct_sum.decomposition",
"linear_algebra.eigenspace.minpoly"
] | [
"is_R_or_C.conj_eq_iff_re"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_eigenvalue_eigenvalues (i : fin n) : has_eigenvalue T (hT.eigenvalues hn i) | module.End.has_eigenvalue_of_has_eigenvector (hT.has_eigenvector_eigenvector_basis hn i) | lemma | linear_map.is_symmetric.has_eigenvalue_eigenvalues | analysis.inner_product_space | src/analysis/inner_product_space/spectrum.lean | [
"analysis.inner_product_space.rayleigh",
"analysis.inner_product_space.pi_L2",
"algebra.direct_sum.decomposition",
"linear_algebra.eigenspace.minpoly"
] | [
"module.End.has_eigenvalue_of_has_eigenvector"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_eigenvector_basis (i : fin n) :
T (hT.eigenvector_basis hn i) = (hT.eigenvalues hn i : 𝕜) • hT.eigenvector_basis hn i | mem_eigenspace_iff.mp (hT.has_eigenvector_eigenvector_basis hn i).1 | lemma | linear_map.is_symmetric.apply_eigenvector_basis | analysis.inner_product_space | src/analysis/inner_product_space/spectrum.lean | [
"analysis.inner_product_space.rayleigh",
"analysis.inner_product_space.pi_L2",
"algebra.direct_sum.decomposition",
"linear_algebra.eigenspace.minpoly"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diagonalization_basis_apply_self_apply (v : E) (i : fin n) :
(hT.eigenvector_basis hn).repr (T v) i =
hT.eigenvalues hn i * (hT.eigenvector_basis hn).repr v i | begin
suffices : ∀ w : euclidean_space 𝕜 (fin n),
T ((hT.eigenvector_basis hn).repr.symm w)
= (hT.eigenvector_basis hn).repr.symm (λ i, hT.eigenvalues hn i * w i),
{ simpa [orthonormal_basis.sum_repr_symm] using
congr_arg (λ v, (hT.eigenvector_basis hn).repr v i)
(this ((hT.eigenvector_basis ... | lemma | linear_map.is_symmetric.diagonalization_basis_apply_self_apply | analysis.inner_product_space | src/analysis/inner_product_space/spectrum.lean | [
"analysis.inner_product_space.rayleigh",
"analysis.inner_product_space.pi_L2",
"algebra.direct_sum.decomposition",
"linear_algebra.eigenspace.minpoly"
] | [
"euclidean_space",
"linear_map.map_smul",
"linear_map.map_sum",
"mul_comm",
"orthonormal_basis.sum_repr_symm",
"smul_smul"
] | *Diagonalization theorem*, *spectral theorem*; version 2: A self-adjoint operator `T` on a
finite-dimensional inner product space `E` acts diagonally on the identification of `E` with
Euclidean space induced by an orthonormal basis of eigenvectors of `T`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inner_product_apply_eigenvector {μ : 𝕜} {v : E} {T : E →ₗ[𝕜] E}
(h : v ∈ module.End.eigenspace T μ) : ⟪v, T v⟫ = μ * ‖v‖ ^ 2 | by simp only [mem_eigenspace_iff.mp h, inner_smul_right, inner_self_eq_norm_sq_to_K] | lemma | inner_product_apply_eigenvector | analysis.inner_product_space | src/analysis/inner_product_space/spectrum.lean | [
"analysis.inner_product_space.rayleigh",
"analysis.inner_product_space.pi_L2",
"algebra.direct_sum.decomposition",
"linear_algebra.eigenspace.minpoly"
] | [
"inner_self_eq_norm_sq_to_K",
"inner_smul_right",
"module.End.eigenspace"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eigenvalue_nonneg_of_nonneg {μ : ℝ} {T : E →ₗ[𝕜] E} (hμ : has_eigenvalue T μ)
(hnn : ∀ (x : E), 0 ≤ is_R_or_C.re ⟪x, T x⟫) : 0 ≤ μ | begin
obtain ⟨v, hv⟩ := hμ.exists_has_eigenvector,
have hpos : 0 < ‖v‖ ^ 2, by simpa only [sq_pos_iff, norm_ne_zero_iff] using hv.2,
have : is_R_or_C.re ⟪v, T v⟫ = μ * ‖v‖ ^ 2,
{ exact_mod_cast congr_arg is_R_or_C.re (inner_product_apply_eigenvector hv.1) },
exact (zero_le_mul_right hpos).mp (this ▸ hnn v),
e... | lemma | eigenvalue_nonneg_of_nonneg | analysis.inner_product_space | src/analysis/inner_product_space/spectrum.lean | [
"analysis.inner_product_space.rayleigh",
"analysis.inner_product_space.pi_L2",
"algebra.direct_sum.decomposition",
"linear_algebra.eigenspace.minpoly"
] | [
"inner_product_apply_eigenvector",
"sq_pos_iff",
"zero_le_mul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eigenvalue_pos_of_pos {μ : ℝ} {T : E →ₗ[𝕜] E} (hμ : has_eigenvalue T μ)
(hnn : ∀ (x : E), 0 < is_R_or_C.re ⟪x, T x⟫) : 0 < μ | begin
obtain ⟨v, hv⟩ := hμ.exists_has_eigenvector,
have hpos : 0 < ‖v‖ ^ 2, by simpa only [sq_pos_iff, norm_ne_zero_iff] using hv.2,
have : is_R_or_C.re ⟪v, T v⟫ = μ * ‖v‖ ^ 2,
{ exact_mod_cast congr_arg is_R_or_C.re (inner_product_apply_eigenvector hv.1) },
exact (zero_lt_mul_right hpos).mp (this ▸ hnn v),
e... | lemma | eigenvalue_pos_of_pos | analysis.inner_product_space | src/analysis/inner_product_space/spectrum.lean | [
"analysis.inner_product_space.rayleigh",
"analysis.inner_product_space.pi_L2",
"algebra.direct_sum.decomposition",
"linear_algebra.eigenspace.minpoly"
] | [
"inner_product_apply_eigenvector",
"sq_pos_iff",
"zero_lt_mul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_symmetric (T : E →ₗ[𝕜] E) : Prop | ∀ x y, ⟪T x, y⟫ = ⟪x, T y⟫ | def | linear_map.is_symmetric | analysis.inner_product_space | src/analysis/inner_product_space/symmetric.lean | [
"analysis.inner_product_space.basic",
"analysis.normed_space.banach",
"linear_algebra.sesquilinear_form"
] | [] | A (not necessarily bounded) operator on an inner product space is symmetric, if for all
`x`, `y`, we have `⟪T x, y⟫ = ⟪x, T y⟫`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_symmetric_iff_sesq_form (T : E →ₗ[𝕜] E) :
T.is_symmetric ↔
@linear_map.is_self_adjoint 𝕜 E _ _ _ (star_ring_end 𝕜) sesq_form_of_inner T | ⟨λ h x y, (h y x).symm, λ h x y, (h y x).symm⟩ | lemma | linear_map.is_symmetric_iff_sesq_form | analysis.inner_product_space | src/analysis/inner_product_space/symmetric.lean | [
"analysis.inner_product_space.basic",
"analysis.normed_space.banach",
"linear_algebra.sesquilinear_form"
] | [
"linear_map.is_self_adjoint",
"sesq_form_of_inner",
"star_ring_end"
] | An operator `T` on an inner product space is symmetric if and only if it is
`linear_map.is_self_adjoint` with respect to the sesquilinear form given by the inner product. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_symmetric.conj_inner_sym {T : E →ₗ[𝕜] E} (hT : is_symmetric T) (x y : E) :
conj ⟪T x, y⟫ = ⟪T y, x⟫ | by rw [hT x y, inner_conj_symm] | lemma | linear_map.is_symmetric.conj_inner_sym | analysis.inner_product_space | src/analysis/inner_product_space/symmetric.lean | [
"analysis.inner_product_space.basic",
"analysis.normed_space.banach",
"linear_algebra.sesquilinear_form"
] | [
"inner_conj_symm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_symmetric.apply_clm {T : E →L[𝕜] E} (hT : is_symmetric (T : E →ₗ[𝕜] E))
(x y : E) : ⟪T x, y⟫ = ⟪x, T y⟫ | hT x y | lemma | linear_map.is_symmetric.apply_clm | analysis.inner_product_space | src/analysis/inner_product_space/symmetric.lean | [
"analysis.inner_product_space.basic",
"analysis.normed_space.banach",
"linear_algebra.sesquilinear_form"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_symmetric_zero : (0 : E →ₗ[𝕜] E).is_symmetric | λ x y, (inner_zero_right x : ⟪x, 0⟫ = 0).symm ▸ (inner_zero_left y : ⟪0, y⟫ = 0) | lemma | linear_map.is_symmetric_zero | analysis.inner_product_space | src/analysis/inner_product_space/symmetric.lean | [
"analysis.inner_product_space.basic",
"analysis.normed_space.banach",
"linear_algebra.sesquilinear_form"
] | [
"inner_zero_left",
"inner_zero_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_symmetric_id : (linear_map.id : E →ₗ[𝕜] E).is_symmetric | λ x y, rfl | lemma | linear_map.is_symmetric_id | analysis.inner_product_space | src/analysis/inner_product_space/symmetric.lean | [
"analysis.inner_product_space.basic",
"analysis.normed_space.banach",
"linear_algebra.sesquilinear_form"
] | [
"linear_map.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_symmetric.add {T S : E →ₗ[𝕜] E} (hT : T.is_symmetric) (hS : S.is_symmetric) :
(T + S).is_symmetric | begin
intros x y,
rw [linear_map.add_apply, inner_add_left, hT x y, hS x y, ← inner_add_right],
refl
end | lemma | linear_map.is_symmetric.add | analysis.inner_product_space | src/analysis/inner_product_space/symmetric.lean | [
"analysis.inner_product_space.basic",
"analysis.normed_space.banach",
"linear_algebra.sesquilinear_form"
] | [
"inner_add_left",
"inner_add_right",
"linear_map.add_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_symmetric.continuous [complete_space E] {T : E →ₗ[𝕜] E} (hT : is_symmetric T) :
continuous T | begin
-- We prove it by using the closed graph theorem
refine T.continuous_of_seq_closed_graph (λ u x y hu hTu, _),
rw [←sub_eq_zero, ←@inner_self_eq_zero 𝕜],
have hlhs : ∀ k : ℕ, ⟪T (u k) - T x, y - T x⟫ = ⟪u k - x, T (y - T x)⟫ :=
by { intro k, rw [←T.map_sub, hT] },
refine tendsto_nhds_unique ((hTu.sub_... | lemma | linear_map.is_symmetric.continuous | analysis.inner_product_space | src/analysis/inner_product_space/symmetric.lean | [
"analysis.inner_product_space.basic",
"analysis.normed_space.banach",
"linear_algebra.sesquilinear_form"
] | [
"complete_space",
"continuous",
"filter.tendsto.inner",
"inner_self_eq_zero",
"tendsto_const_nhds",
"tendsto_nhds_unique"
] | The **Hellinger--Toeplitz theorem**: if a symmetric operator is defined on a complete space,
then it is automatically continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_symmetric.coe_re_apply_inner_self_apply
{T : E →L[𝕜] E} (hT : is_symmetric (T : E →ₗ[𝕜] E)) (x : E) :
(T.re_apply_inner_self x : 𝕜) = ⟪T x, x⟫ | begin
rsuffices ⟨r, hr⟩ : ∃ r : ℝ, ⟪T x, x⟫ = r,
{ simp [hr, T.re_apply_inner_self_apply] },
rw ← conj_eq_iff_real,
exact hT.conj_inner_sym x x
end | lemma | linear_map.is_symmetric.coe_re_apply_inner_self_apply | analysis.inner_product_space | src/analysis/inner_product_space/symmetric.lean | [
"analysis.inner_product_space.basic",
"analysis.normed_space.banach",
"linear_algebra.sesquilinear_form"
] | [] | For a symmetric operator `T`, the function `λ x, ⟪T x, x⟫` is real-valued. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_symmetric.restrict_invariant {T : E →ₗ[𝕜] E} (hT : is_symmetric T)
{V : submodule 𝕜 E} (hV : ∀ v ∈ V, T v ∈ V) :
is_symmetric (T.restrict hV) | λ v w, hT v w | lemma | linear_map.is_symmetric.restrict_invariant | analysis.inner_product_space | src/analysis/inner_product_space/symmetric.lean | [
"analysis.inner_product_space.basic",
"analysis.normed_space.banach",
"linear_algebra.sesquilinear_form"
] | [
"submodule"
] | If a symmetric operator preserves a submodule, its restriction to that submodule is
symmetric. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_symmetric.restrict_scalars {T : E →ₗ[𝕜] E} (hT : T.is_symmetric) :
@linear_map.is_symmetric ℝ E _ _ (inner_product_space.is_R_or_C_to_real 𝕜 E)
(@linear_map.restrict_scalars ℝ 𝕜 _ _ _ _ _ _
(inner_product_space.is_R_or_C_to_real 𝕜 E).to_module
(inner_product_space.is_R_or_C_to_real 𝕜 E).to_module _ ... | λ x y, by simp [hT x y, real_inner_eq_re_inner, linear_map.coe_restrict_scalars_eq_coe] | lemma | linear_map.is_symmetric.restrict_scalars | analysis.inner_product_space | src/analysis/inner_product_space/symmetric.lean | [
"analysis.inner_product_space.basic",
"analysis.normed_space.banach",
"linear_algebra.sesquilinear_form"
] | [
"inner_product_space.is_R_or_C_to_real",
"linear_map.coe_restrict_scalars_eq_coe",
"linear_map.is_symmetric",
"linear_map.restrict_scalars",
"real_inner_eq_re_inner"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_symmetric_iff_inner_map_self_real (T : V →ₗ[ℂ] V):
is_symmetric T ↔ ∀ (v : V), conj ⟪T v, v⟫_ℂ = ⟪T v, v⟫_ℂ | begin
split,
{ intros hT v,
apply is_symmetric.conj_inner_sym hT },
{ intros h x y,
nth_rewrite 1 ← inner_conj_symm,
nth_rewrite 1 inner_map_polarization,
simp only [star_ring_end_apply, star_div', star_sub, star_add, star_mul],
simp only [← star_ring_end_apply],
rw [h (x + y), h (x - y), ... | lemma | linear_map.is_symmetric_iff_inner_map_self_real | analysis.inner_product_space | src/analysis/inner_product_space/symmetric.lean | [
"analysis.inner_product_space.basic",
"analysis.normed_space.banach",
"linear_algebra.sesquilinear_form"
] | [
"complex.I",
"complex.conj_I",
"inner_conj_symm",
"inner_map_polarization",
"inner_map_polarization'",
"ring",
"star_div'",
"star_ring_end_apply",
"star_sub"
] | A linear operator on a complex inner product space is symmetric precisely when
`⟪T v, v⟫_ℂ` is real for all v. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_symmetric.inner_map_polarization {T : E →ₗ[𝕜] E} (hT : T.is_symmetric) (x y : E) :
⟪T x, y⟫ = (⟪T (x + y), x + y⟫ - ⟪T (x - y), x - y⟫ -
I * ⟪T (x + (I : 𝕜) • y), x + (I : 𝕜) • y⟫ +
I * ⟪T (x - (I : 𝕜) • y), x - (I : 𝕜) • y⟫) / 4 | begin
rcases @I_mul_I_ax 𝕜 _ with (h | h),
{ simp_rw [h, zero_mul, sub_zero, add_zero, map_add, map_sub, inner_add_left,
inner_add_right, inner_sub_left, inner_sub_right, hT x, ← inner_conj_symm x (T y)],
suffices : (re ⟪T y, x⟫ : 𝕜) = ⟪T y, x⟫,
{ rw conj_eq_iff_re.mpr this,
ring, },
{ rw ... | lemma | linear_map.is_symmetric.inner_map_polarization | analysis.inner_product_space | src/analysis/inner_product_space/symmetric.lean | [
"analysis.inner_product_space.basic",
"analysis.normed_space.banach",
"linear_algebra.sesquilinear_form"
] | [
"inner_add_left",
"inner_add_right",
"inner_conj_symm",
"inner_smul_left",
"inner_smul_right",
"inner_sub_left",
"inner_sub_right",
"is_R_or_C.conj_I",
"linear_map.map_smul",
"mul_assoc",
"mul_neg",
"mul_zero",
"neg_one_mul",
"one_mul",
"ring",
"zero_mul"
] | Polarization identity for symmetric linear maps.
See `inner_map_polarization` for the complex version without the symmetric assumption. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_symmetric.inner_map_self_eq_zero {T : E →ₗ[𝕜] E} (hT : T.is_symmetric) :
(∀ x, ⟪T x, x⟫ = 0) ↔ T = 0 | begin
simp_rw [linear_map.ext_iff, zero_apply],
refine ⟨λ h x, _, λ h, by simp_rw [h, inner_zero_left, forall_const]⟩,
rw [← @inner_self_eq_zero 𝕜, hT.inner_map_polarization],
simp_rw [h _],
ring,
end | lemma | linear_map.is_symmetric.inner_map_self_eq_zero | analysis.inner_product_space | src/analysis/inner_product_space/symmetric.lean | [
"analysis.inner_product_space.basic",
"analysis.normed_space.banach",
"linear_algebra.sesquilinear_form"
] | [
"forall_const",
"inner_self_eq_zero",
"inner_zero_left",
"linear_map.ext_iff",
"ring"
] | A symmetric linear map `T` is zero if and only if `⟪T x, x⟫_ℝ = 0` for all `x`.
See `inner_map_self_eq_zero` for the complex version without the symmetric assumption. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
area_form : E →ₗ[ℝ] E →ₗ[ℝ] ℝ | begin
let z : alternating_map ℝ E ℝ (fin 0) ≃ₗ[ℝ] ℝ :=
alternating_map.const_linear_equiv_of_is_empty.symm,
let y : alternating_map ℝ E ℝ (fin 1) →ₗ[ℝ] E →ₗ[ℝ] ℝ :=
(linear_map.llcomp ℝ E (alternating_map ℝ E ℝ (fin 0)) ℝ z) ∘ₗ
alternating_map.curry_left_linear_map,
exact y ∘ₗ (alternating_map.curry... | def | orientation.area_form | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [
"alternating_map",
"alternating_map.curry_left_linear_map",
"linear_map.llcomp"
] | An antisymmetric bilinear form on an oriented real inner product space of dimension 2 (usual
notation `ω`). When evaluated on two vectors, it gives the oriented area of the parallelogram they
span. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
area_form_to_volume_form (x y : E) : ω x y = o.volume_form ![x, y] | by simp [area_form] | lemma | orientation.area_form_to_volume_form | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
area_form_apply_self (x : E) : ω x x = 0 | begin
rw area_form_to_volume_form,
refine o.volume_form.map_eq_zero_of_eq ![x, x] _ (_ : (0 : fin 2) ≠ 1),
{ simp },
{ norm_num }
end | lemma | orientation.area_form_apply_self | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
area_form_swap (x y : E) : ω x y = - ω y x | begin
simp only [area_form_to_volume_form],
convert o.volume_form.map_swap ![y, x] (_ : (0 : fin 2) ≠ 1),
{ ext i,
fin_cases i; refl },
{ norm_num }
end | lemma | orientation.area_form_swap | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
area_form_neg_orientation : (-o).area_form = -o.area_form | begin
ext x y,
simp [area_form_to_volume_form]
end | lemma | orientation.area_form_neg_orientation | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
area_form' : E →L[ℝ] (E →L[ℝ] ℝ) | ((↑(linear_map.to_continuous_linear_map : (E →ₗ[ℝ] ℝ) ≃ₗ[ℝ] (E →L[ℝ] ℝ)))
∘ₗ o.area_form).to_continuous_linear_map | def | orientation.area_form' | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [
"linear_map.to_continuous_linear_map"
] | Continuous linear map version of `orientation.area_form`, useful for calculus. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
area_form'_apply (x : E) :
o.area_form' x = (o.area_form x).to_continuous_linear_map | rfl | lemma | orientation.area_form'_apply | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_area_form_le (x y : E) : |ω x y| ≤ ‖x‖ * ‖y‖ | by simpa [area_form_to_volume_form, fin.prod_univ_succ] using o.abs_volume_form_apply_le ![x, y] | lemma | orientation.abs_area_form_le | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [
"fin.prod_univ_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
area_form_le (x y : E) : ω x y ≤ ‖x‖ * ‖y‖ | by simpa [area_form_to_volume_form, fin.prod_univ_succ] using o.volume_form_apply_le ![x, y] | lemma | orientation.area_form_le | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [
"fin.prod_univ_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_area_form_of_orthogonal {x y : E} (h : ⟪x, y⟫ = 0) : |ω x y| = ‖x‖ * ‖y‖ | begin
rw [o.area_form_to_volume_form, o.abs_volume_form_apply_of_pairwise_orthogonal],
{ simp [fin.prod_univ_succ] },
intros i j hij,
fin_cases i; fin_cases j,
{ simpa },
{ simpa using h },
{ simpa [real_inner_comm] using h },
{ simpa }
end | lemma | orientation.abs_area_form_of_orthogonal | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [
"fin.prod_univ_succ",
"real_inner_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
area_form_map {F : Type*}
[normed_add_comm_group F] [inner_product_space ℝ F] [fact (finrank ℝ F = 2)]
(φ : E ≃ₗᵢ[ℝ] F) (x y : F) :
(orientation.map (fin 2) φ.to_linear_equiv o).area_form x y = o.area_form (φ.symm x) (φ.symm y) | begin
have : φ.symm ∘ ![x, y] = ![φ.symm x, φ.symm y],
{ ext i,
fin_cases i; refl },
simp [area_form_to_volume_form, volume_form_map, this],
end | lemma | orientation.area_form_map | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [
"fact",
"inner_product_space",
"normed_add_comm_group",
"orientation.map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
area_form_comp_linear_isometry_equiv (φ : E ≃ₗᵢ[ℝ] E)
(hφ : 0 < (φ.to_linear_equiv : E →ₗ[ℝ] E).det) (x y : E) :
o.area_form (φ x) (φ y) = o.area_form x y | begin
convert o.area_form_map φ (φ x) (φ y),
{ symmetry,
rwa ← o.map_eq_iff_det_pos φ.to_linear_equiv at hφ,
rw [fact.out (finrank ℝ E = 2), fintype.card_fin] },
{ simp },
{ simp }
end | lemma | orientation.area_form_comp_linear_isometry_equiv | analysis.inner_product_space | src/analysis/inner_product_space/two_dim.lean | [
"analysis.inner_product_space.dual",
"analysis.inner_product_space.orientation",
"data.complex.orientation",
"tactic.linear_combination"
] | [
"fintype.card_fin"
] | The area form is invariant under pullback by a positively-oriented isometric automorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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