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Proposition 17.2 Every quadratic form \( q \) of rank \( r \) over \( {\mathbb{F}}_{p}, p > 2 \), is equivalent to \( {x}_{1}^{2} + \cdots + {x}_{r - 1}^{2} + {x}_{r}^{2} \) if \( \det {q}_{\text{red }} \in {\mathbb{F}}_{p}^{2} \), and is equivalent to \( {x}_{1}^{2} + \cdots + {x}_{r - 1}^{2} + \varepsilon {x}_{r}^{2}... | Proof By Lagrange's theorem, Theorem 16.4 on p. 409,\n\n\[ q\left( x\right) = {\alpha }_{1}{x}_{1}^{2} + {\alpha }_{2}{x}_{2}^{2} + \cdots + {\alpha }_{r}{x}_{r}^{2} \]\n\nin appropriate coordinates. It is enough to show that every linear combination of two squares \( {\alpha }_{i}{x}_{i}^{2} + {\alpha }_{j}{x}_{j}^{2}... | Yes |
Proposition 17.3 Up to isometry, there are exactly three anisotropic forms over \( {\mathbb{F}}_{p} \) with \( p > 2 \), namely \( {x}_{1}^{2},\varepsilon {x}_{1}^{2} \), and either \( {x}_{1}^{2} + {x}_{2}^{2} \), for \( p \equiv - 1\\left( {\\;\\operatorname{mod}\\;4}\\right) \), or \( {x}_{1}^{2} + \\varepsilon {x}_... | Proof By Lemma 17.3, every form \( a{x}_{1}^{2} + b{x}_{2}^{2} + c{x}_{3}^{2} + \\cdots \) of rank \( \\geq 3 \) vanishes on a nonzero vector \( \\left( {{\\alpha }_{1},{\\alpha }_{2},1,0,\\ldots }\\right) \) such that \( a{\\alpha }_{1}^{2} + b{\\alpha }_{2}^{2} = - c \) . Therefore, anisotropic forms over \( {\\mathb... | Yes |
Lemma 17.4 A line (ab) is tangent to a quadric \( Q = Z\left( q\right) \) at a point \( a \in Q \) if and only if \( \widetilde{q}\left( {a, b}\right) = 0 \) . | Proof Write \( U \subset V \) for the liner span of the vectors \( a, b \in V \) . The restriction \( {\left. q\right| }_{U} \) is either zero or singular if and only if its Gramian \( G = \left( \begin{matrix} 0 & \widetilde{q}\left( {a, b}\right) \\ \widetilde{q}\left( {a, b}\right) & \widetilde{q}\left( {b, b}\right... | Yes |
Corollary 17.6 If a point \( a \in Q = Z\left( q\right) \subset \mathbb{P}\left( V\right) \) is smooth, then\n\n\[ \n{T}_{a}Q = \left\{ {x \in {\mathbb{P}}_{n} \mid \widetilde{q}\left( {a, x}\right) = 0}\right\} \n\]\n\nis a hyperplane. If \( a \in \operatorname{Sing}Q \), then \( {T}_{a}Q = \mathbb{P}\left( V\right) \... | Proof The linear equation \( \widetilde{q}\left( {a, x}\right) = 0 \) either defines a hyperplane or is satisfied identically in \( x \) . The latter means that \( a \in \ker q \). | Yes |
Corollary 17.7 (Apparent Contour) The apparent contour of a quadric \( Q = Z\left( q\right) \) viewed from a point \( {}^{29}b \notin Q \) is cut out of \( Q \) by the hyperplane\n\n\[ {b}^{ \bot } = \operatorname{Ann}\widehat{q}\left( b\right) = \{ x \mid \widetilde{q}\left( {b, x}\right) = 0\} . \] | Proof If \( b \notin Q \), then \( \widetilde{q}\left( {b, b}\right) = q\left( b\right) \neq 0 \) . Therefore, the linear equation \( \widetilde{q}\left( {b, x}\right) = 0 \) is nontrivial and defines a hyperplane. | Yes |
Proposition 17.5 If a quadric \( Q \subset {\mathbb{P}}_{n} \) has a smooth point \( a \in Q \), then \( Q \) is not contained in a hyperplane. | Proof For \( n = 1 \), this follows from Example 17.3. Consider \( n \geq 2 \) . If \( Q \) lies within a hyperplane \( H \), then every line \( \ell ⊄ H \) passing through \( a \) intersects \( Q \) only in \( a \) and therefore is tangent to \( Q \) at \( a \) . Hence, \( {\mathbb{P}}_{n} = H \cup {T}_{p}Q \) . This ... | No |
Theorem 17.3 Let \( Q \subset {\mathbb{P}}_{n} \) be an arbitrary quadric and \( L \subset {\mathbb{P}}_{n} \) a projective subspace complementary \( {}^{30} \) to \( \operatorname{Sing}Q \) . Then \( {Q}^{\prime } = L \cap Q \) is a smooth quadric in \( L \), and \( Q \) is the linear join \( {}^{31} \) of \( {Q}^{\pr... | Proof The smoothness of \( {Q}^{\prime } \) follows from Proposition 16.6 on p. 409. Every line intersecting Sing \( Q \) either belongs to Sing \( Q \) or can be written as (ab) for some \( a \in \operatorname{Sing}Q \) and \( b \in L \) . By Corollary 17.6, the line \( \left( {ab}\right) \) either lies on \( Q \), in... | Yes |
Example 17.5 (Veronese Conic) Consider \( {\mathbb{P}}_{1} = \mathbb{P}\left( U\right) \) and write \( {S}^{2}{\mathbb{P}}_{1} \) for the set of unordered pairs of points \( \{ a, b\} \subset {\mathbb{P}}_{1} \), where \( a = b \) is allowed as well. Over an algebraically closed field, there is canonical bijection\n\n\... | In the basis \( {x}_{0}^{2},2{x}_{0}{x}_{1},{x}_{1}^{2} \) for \( {S}^{2}{U}^{ * } \), where the quadratic form \( {\vartheta }_{0}{x}_{0}^{2} + 2{\vartheta }_{1}{x}_{0}{x}_{1} + {\vartheta }_{2}{x}_{1}^{2} \) has coordinates \( \left( {{\vartheta }_{0} : {\vartheta }_{1} : {\vartheta }_{2}}\right) \), the form \( {q}_... | Yes |
Proposition 17.6 Let \( C \subset {\mathbb{P}}_{2} \) be a smooth conic and \( D \subset {\mathbb{P}}_{2} \) a curve of degree \( d \) . Then either \( C \subset D \) or \( C \) intersects \( D \) in at most \( {2d} \) points. | Proof Let \( D = Z\left( f\right) \) and \( C = Z\left( q\right) \) . If \( C = \varnothing \), there is nothing to prove. If there is some \( a \in C \), choose any line \( \ell ∌ a \) and consider the projection \( {\pi }_{a} : C \rightarrow \ell \) from \( a \) onto \( \ell \) . It is bijective, because each line \(... | Yes |
Example 17.6 (Segre Quadric) Fix a 2-dimensional vector space \( U \), put \( W = \) \( \operatorname{End}\left( U\right) \), and consider \( {\mathbb{P}}_{3} = \mathbb{P}\left( W\right) \) . Write\n\n\[ \n{Q}_{S}\overset{\text{ def }}{ = }\{ F : \in \operatorname{End}\left( U\right) \mid \det F = 0\} = \left\{ {\left.... | Every rank-1 operator \( F : U \rightarrow U \) has a 1-dimensional image spanned by some vector \( v \in U \), uniquely determined by \( F \) up to proportionality. Then the value of \( F \) on every \( u \in U \) equals \( F\left( u\right) = \xi \left( u\right) \cdot v \) , where \( \xi \in {U}^{ * } \) is a linear f... | Yes |
Over an algebraically closed field, the only anisotropic form is the 1-dimensional form \( {x}^{2} \) . Hence, for each \( n \in \mathbb{N} \), there exists exactly one smooth \( n \) -dimensional quadric \( {Q}_{n} \subset {\mathbb{P}}_{n + 1} \) up to isomorphism. For even \( n = {2m} \), it can be defined by the equ... | The planarity of both quadrics (17.16),(17.17) is \( m \), that is, some \( m \)-dimensional projective subspace \( L \subset {Q}_{n} \) can be drawn through each point \( a \in {Q}_{n} \), and there are no \( \left( {m + 1}\right) \)-dimensional subspaces lying on \( {Q}_{n} \). | No |
Proposition 17.7 For every smooth quadric \( Q \) and hyperplane \( \Pi \), the intersection \( \Pi \cap Q \subset \Pi \) either is a smooth quadric in \( \Pi \) or has exactly one singular point \( p \) . In the latter case, \( \Pi = {T}_{p}Q \), and the quadric \( \Pi \cap Q \) is a cone with vertex \( p \) over the ... | Proof Let \( Q = Z\left( q\right) \subset \mathbb{P}\left( V\right) \) and \( \Pi = \mathbb{P}\left( W\right) \) . Then\n\n\[ \dim \ker \left( {\widehat{q}{|}_{W}}\right) = \dim \left( {W \cap {\widehat{q}}^{-1}\left( {\operatorname{Ann}W}\right) }\right) \leq \dim {\widehat{q}}^{-1}\left( {\operatorname{Ann}W}\right) ... | Yes |
If \( \mathbb{k} \) is algebraically closed, then the smooth quadrics \( {Q}_{0} \subset {\mathbb{P}}_{1} \) and \( {Q}_{1} \subset {\mathbb{P}}_{2} \) both are 0 -planar. The next two quadrics \( {Q}_{2} \subset {\mathbb{P}}_{3} \) and \( {Q}_{3} \subset {\mathbb{P}}_{4} \) are 1-planar. | For every \( p \in {Q}_{2} \), there are exactly two lines \( \ell \subset {Q}_{2} \) passing through \( p \) . They join \( p \) with two points of some smooth quadric \( {Q}_{0} \subset {\mathbb{P}}_{1} \subset {T}_{p}{Q}_{2} \smallsetminus \{ p\} \) . Note that this agrees with Example 17.6. The lines \( \ell \subse... | Yes |
Proposition 17.8 Let \( Q \) be a smooth quadric and \( a, b \notin Q \) two different points such that the line \( \ell = \left( {ab}\right) \) intersects \( Q \) in two different points \( c, d \) . Then \( a, b \) are conjugate with respect to \( Q \) if and only if the pair \( \{ a, b\} \) is harmonic \( {}^{37} \)... | Proof Let \( Q = Z\left( q\right) \) and write \( x = \left( {{x}_{0} : {x}_{1}}\right) \) for the homogeneous coordinates on \( \ell \) in the basis \( c, d \in \ell \) . In these coordinates, the restriction of \( q \) onto \( \ell \) is given, up to proportionality, by the quadratic form \( q\left( x\right) = \det \... | Yes |
Proposition 17.9 Let \( G \subset {\mathbb{P}}_{n} \) be a smooth quadric with Gramian \( \Gamma \), and \( Q \subset {\mathbb{P}}_{n} \) an arbitrary quadric with Gramian \( B \) in the same basis. Then the polar map \( {\mathbb{P}}_{n} \rightarrow {\mathbb{P}}_{n}^{ \times } \) provided by \( G \) sends \( Q \) to a ... | Proof In coordinates with respect to any dual bases in \( V \) and \( {V}^{ * } \), the correlation \( \widehat{g} : V \simeq {V}^{ * } \) takes a vector with coordinate row \( x \) to the covector with coordinate row \( y = {x\Gamma } \) . If the vector \( x \) is constrained by the relation \( {xB}{x}^{t} = 0 \), the... | Yes |
Corollary 17.9 The tangent spaces of a smooth quadric \( S \subset {\mathbb{P}}_{n} \) form a smooth quadric \( {S}^{ \times } \subset {\mathbb{P}}_{n}^{ \times } \) . The Gramians of \( S \) and \( {S}^{ \times } \) in dual bases of \( {\mathbb{P}}_{n} \) and \( {\mathbb{P}}_{n}^{ \times } \) are inverse to each other... | Proof Apply Proposition 17.9 for \( G = Q = S \) . | No |
Proposition 17.10 Over an infinite field, two nonempty smooth quadrics coincide if and only if their equations are proportional. | Proof Let \( Z\left( {q}_{1}\right) = Z\left( {q}_{2}\right) \) in \( \mathbb{P}\left( V\right) \) . Then the two polarities \( {\bar{q}}_{1},{\bar{q}}_{2} : \mathbb{P}\left( V\right) \simeq \mathbb{P}\left( {V}^{ * }\right) \) coincide at all points of the quadrics. | No |
Proposition 17.11 Let two nonempty affine quadrics \( {X}^{\prime } = Z\left( {f}^{\prime }\right) ,{X}^{\prime \prime } = Z\left( {f}^{\prime \prime }\right) \subset \) \( \mathbb{A}\left( V\right) \) have projective closures \( {Q}^{\prime },{Q}^{\prime \prime } \subset \mathbb{P}\left( W\right) \) . Then \( {X}^{\pr... | Proof We identify \( {X}^{\prime } \) and \( {X}^{\prime \prime } \) with \( {Q}_{\text{aff }}^{\prime } = {Q}^{\prime } \cap {U}_{0} \) and \( {Q}_{\text{aff }}^{\prime \prime } = {Q}^{\prime \prime } \cap {U}_{0} \) respectively. Let us show first that an affine automorphism \( \varphi : {U}_{0} \simeq {U}_{0} \) is ... | Yes |
Lemma 17.5 Let \( H \subset {\mathbb{P}}_{n} \) be a hyperplane and \( Q \subset {\mathbb{P}}_{n} \) a nonempty quadric such that \( Q ⊄ H \) and \( H ⊄ Q \) . If the ground field is infinite, then \( Q \cap H \) is uniquely determined by \( Q \smallsetminus H \) . | Proof For \( n = 1 \), the statement is obvious from Corollary 17.5 on p. 435 . Consider \( n \geq 2 \) . If \( Q = V\left( q\right) \) is smooth, then the same arguments as in Proposition 17.10 on p. 444 allow us to find \( n + 2 \) points in \( Q \smallsetminus H \) such that no \( n + 1 \) of them lie within a hyper... | Yes |
Problem 17.4 Expand the characteristic polynomial of the matrix \( X \in {\operatorname{Mat}}_{n}\left( \mathbb{R}\right) \) as\n\n\[ \det \left( {{tE} - X}\right) = {t}^{n} - {\sigma }_{1}\left( X\right) {t}^{n - 1} + {\sigma }_{2}\left( X\right) {t}^{n - 2} + \cdots + {\left( -1\right) }^{n}\det X. \] | Convince yourself that \( {\sigma }_{2}\left( X\right) \) is a quadratic form on the vector space \( {\operatorname{Mat}}_{n}\left( \mathbb{R}\right) \) and find its rank and signature. To begin with, consider \( n = 2,3,4 \) . | No |
Proposition 18.1 (Cauchy-Riemann Relations) An operator \( G \in {\operatorname{End}}_{\mathbb{R}}\left( {W}_{\mathbb{R}}\right) \) with matrix (18.2) in the basis (18.1) lies in the subalgebra \( {\operatorname{End}}_{\mathbb{C}}\left( W\right) \) of \( {\operatorname{End}}_{\mathbb{R}}\left( {W}_{\mathbb{R}}\right) \... | Proof The \( \mathbb{C} \) -linearity of \( G \) means that \( G\left( {iw}\right) = {iG}\left( w\right) \) for all \( w \in {W}_{\mathbb{R}} \) . Since this relation is \( \mathbb{R} \) -linear in \( w \), it is enough to check it for the basis vectors (18.1) only. This forces \( C = B \) and \( D = - A \) . Conversel... | Yes |
A function \( f : \mathbb{C} \rightarrow \mathbb{C} \) , \( z \mapsto w = f\left( z\right) \), is called complex-differentiable at a point \( {z}_{0} = {x}_{0} + i{y}_{0} \) if its increment is approximated by the \( \mathbb{C} \) -linear function of the increment of its argument, that is, if\n\n\[ f\left( {{z}_{0} + {... | It is not hard to check \( {}^{1} \) that if the approximations (18.4) and (18.5) exist, then both linear operators \( {}^{2} \) acting on the increments of the arguments are expressed through the derivatives of the functions in question as follows:\n\n\[ \zeta = \frac{df}{dz}\left( {z}_{0}\right) = \mathop{\lim }\limi... | No |
Proposition 18.2 Every elementary divisor \( {\left( t - \lambda \right) }^{m} \in \mathcal{E}\ell \left( F\right) \), where \( \lambda \in \mathbb{R} \), is simultaneously an elementary divisor for \( {F}_{\mathbb{C}} \) . Every elementary divisor \( {p}^{m}\left( t\right) \in \) \( \mathcal{E}\ell \left( F\right) \) ... | Proof The complexification of the real vector space \( \mathbb{R}\left\lbrack t\right\rbrack /\left( {p}^{m}\right) \) is the complex vector space \( \mathbb{C}\left\lbrack t\right\rbrack /\left( {p}^{m}\right) \) . The multiplication-by- \( t \) operator in the first space is complexified to the multiplication-by- \( ... | Yes |
Proposition 18.3 For every vector space \( V \) over \( \mathbb{R} \), the following data are in canonical bijection:\n\n(1) multiplication of vectors by complex numbers \( \mathbb{C} \times V \rightarrow V \) that makes a vector space over \( \mathbb{C} \) from \( V \) and forces \( V \) to be the realification of thi... | The correspondence \( \left( 1\right) \rightarrow \left( 2\right) \) sends multiplication \( \mathbb{C} \times V \rightarrow V \) to the multiplication-by-i operator \( I : v \mapsto {iv} \). The correspondence \( \left( 2\right) \rightarrow \left( 3\right) \) sends the operator \( I \) to the \( \left( {+i}\right) \) ... | Yes |
Example 18.4 (Space of Integrable Functions) The infinite-dimensional analogue of (18.14) is the standard Hermitian structure on the space of continuous functions \( \left\lbrack {a, b}\right\rbrack \rightarrow \mathbb{C} \) defined by\n\n\[ \left( {f, g}\right) = {\int }_{a}^{b}f\left( x\right) \overline{g\left( x\rig... | where the integral of a complex-valued function \( h\left( x\right) = u\left( x\right) + {iv}\left( x\right) \) with \( u, v \) : \( \left\lbrack {a, b}\right\rbrack \rightarrow \mathbb{R} \) is defined as\n\n\[ {\int }_{a}^{b}{hdx} = {\int }_{a}^{b}\left( {u\left( x\right) + {iv}\left( x\right) }\right) {dx}\overset{\... | Yes |
Example 18.5 (Hermitian Complexification of Euclidean Space) Let \( V \) be a real Euclidean vector space with inner product \( \left( {*, * }\right) : V \times V \rightarrow \mathbb{R} \) . Then the standard Hermitian extension of this product to the complexified space \( W = {V}_{\mathbb{C}} \) is given by the follow... | \[ {\left( {u}_{1} + i{v}_{1},{u}_{2} + i{v}_{2}\right) }_{\mathrm{H}}\overset{\text{ def }}{ = }\left( {\left( {{u}_{1},{u}_{2}}\right) + \left( {{v}_{1},{v}_{2}}\right) }\right) + i\left( {\left( {{u}_{1},{v}_{2}}\right) - \left( {{v}_{1},{u}_{2}}\right) }\right) . \] | Yes |
Proposition 18.4 (Hermitian Completions of a Given Euclidean Structure) For every (even-dimensional) real Euclidean vector space \( \\left( {V, g}\\right) \), the following data are in canonical bijection:\n\n(1) the Kähler triple \( \\left( {I, g,\\omega }\\right) \),\n\n(2) the Euclidean isometry \( I \\in {\\mathrm{... | Proof As we have seen above, every complex structure \( I : V \\rightarrow V \) extending \( g \) to a Kähler triple must be \( g \) -orthogonal. Conversely, given some complex structure \( I \\in {\\mathrm{O}}_{g}\\left( V\\right) \), the \( \\mathbb{R} \) -bilinear form \( g\\left( {v,{Iw}}\\right) \) is nondegenerat... | Yes |
Theorem 18.1 The Kähler triples \( \left( {I, g,\omega }\right) \) completing the standard symplectic structure \( \omega \) in \( {\mathbb{R}}^{2n} \) to a Hermitian structure are in bijection with the symmetric complex matrices \( S \in {\operatorname{Mat}}_{n}\left( \mathbb{C}\right) \) having positive imaginary par... | Proof Only formula (18.21) remains to be verified. By Proposition 18.3, the complex structure \( I : V \rightarrow V \) coming from the decomposition \( {V}_{\mathbb{C}} = U \oplus \bar{U} \) sends the vector \( v = \operatorname{Re}w \in V \) to \( I\left( v\right) = \operatorname{Re}\left( {iw}\right) \) for every \(... | Yes |
Show that \( {\operatorname{End}}_{\mathbb{R}}\left( {W}_{\mathbb{R}}\right) = {\operatorname{End}}_{\mathbb{C}}\left( {W}_{\mathbb{R}}\right) \oplus {\operatorname{End}}_{\overline{\mathbb{C}}}\left( {W}_{\mathbb{R}}\right) \) and that every \( F \in {\operatorname{End}}_{\mathbb{R}}\left( {W}_{\mathbb{R}}\right) \) c... | Also show that the map \( F \mapsto {IF} \) provides both spaces \( {\operatorname{End}}_{\mathbb{C}}\left( {W}_{\mathbb{R}}\right) \) and \( {\operatorname{End}}_{\overline{\mathbb{C}}}\left( {W}_{\mathbb{R}}\right) \) with complex structures in which they become a pair of conjugate complex vector spaces in the sense ... | No |
Problem 18.10 Show that the Siegel upper half-space \( {\mathfrak{H}}_{n} \subset {\operatorname{Mat}}_{n}\left( \mathbb{C}\right) \) is contractable. | That is, there exists a continuous map \( \gamma : {\mathfrak{H}}_{n} \times \left\lbrack {0,1}\right\rbrack \rightarrow {\mathfrak{H}}_{n} \) whose restrictions to \( {\mathfrak{H}}_{n} \times \{ 0\} \) and to \( {\mathfrak{H}}_{n} \times \{ 1\} \) are, respectively, the identity map \( {\operatorname{Id}}_{{\mathfrak... | No |
Lemma 19.1 For every collection of vectors \( w = \left( {{w}_{1},{w}_{2},\ldots ,{w}_{m}}\right) \), the Gram determinant \( {\Gamma }_{w} = \det {G}_{w} \) is a real nonnegative number vanishing if and only if the vectors are linearly related. | Proof Let \( w = e{C}_{ew} \), where \( e = \left( {{e}_{1},{e}_{2},\ldots ,{e}_{n}}\right) \) is an orthonormal basis in the linear span of \( w \) . Then \( {G}_{w} = {C}_{ew}^{t}{\bar{C}}_{ew} \) . If \( n < m \), then \( \operatorname{rk}{G}_{w} \leq \operatorname{rk}{C}_{ew} \leq n < m \) . This forces \( \det {G}... | Yes |
Corollary 19.1 (Triangle Inequality) \( \parallel u\parallel + \parallel w\parallel \geq \parallel u + w\parallel \) for all \( u, w \in W \) . | Proof \( \parallel u + w{\parallel }^{2} = \parallel u{\parallel }^{2} + \parallel w{\parallel }^{2} + 2\left| \left( {u, w}\right) \right| \leq \parallel u{\parallel }^{2} + \parallel w{\parallel }^{2} + 2\parallel u\parallel \cdot \parallel w\parallel = \) \( {\left( \parallel u\parallel + \parallel w\parallel \right... | Yes |
Proposition 19.1 For every \( w \in W \), there exists a unique \( {w}_{U} \in U \) with the following equivalent properties:\n\n(1) \( w - {w}_{U} \in {U}^{ \bot } \), \n\n(2) \( \left( {w, u}\right) = \left( {{w}_{U}, u}\right) \forall u \in U \), \n\n(3) \( \begin{Vmatrix}{w - {v}_{U}}\end{Vmatrix} < \parallel w - u... | For every pair of Hermitian dual bases \( {u}_{1},{u}_{2},\ldots ,{u}_{k} \) and \( {u}^{ \vee }{}_{1},{u}^{ \vee }{}_{2},\ldots ,{u}^{ \vee }{}_{k} \) in \( U \), \n\n\[ \n{\pi }_{U}w = {w}_{U} = \mathop{\sum }\limits_{{i = 1}}^{k}\left( {w,{u}_{i}^{ \vee }}\right) \cdot {u}_{i}. \n\] \n\n(19.6) \n\nProof Completely t... | No |
Example 19.1 (Adjoint Linear Differential Operators) Write \( V \) for the space of infinitely differentiable functions \( f : \left\lbrack {a, b}\right\rbrack \rightarrow \mathbb{R} \) vanishing together with all derivatives at the endpoints \( a, b \) . Introduce a Euclidean inner product on \( V \) by the prescripti... | \[ \left( {\frac{d}{dt}f, g}\right) = {\int }_{a}^{b}{f}^{\prime }{gdt} = - {\int }_{a}^{b}f{g}^{\prime }{dt} = \left( {f, - \frac{d}{dt}g}\right) . \] | Yes |
Theorem 19.1 An operator \( F \) on a Hermitian space \( W \) is normal if and only if it can be diagonalized in some orthonormal basis of \( W \) . In this case, up to permutations of diagonal elements, the diagonal matrix of \( F \) does not depend on the choice of such an orthonormal basis. | Proof Let the matrix of \( F \) in some orthonormal basis be diagonal. Then the matrix of the adjoint operator \( {F}^{ \dagger } \) is also diagonal in this basis and therefore commutes with the matrix of \( F \) . Since diagonal elements \( \lambda \) of every diagonal matrix of \( F \) are in bijection with the elem... | Yes |
Proposition 19.2 Every self-adjoint operator \( F \) on a real Euclidean space \( V \) can be diagonalized in some orthonormal basis. Up to permutation of the diagonal elements, the result of such a diagonalization does not depend on the choice of orthonormal basis. | Proof Write \( {W}_{\lambda } = \left\{ {w \in {V}_{\mathbb{C}} \mid {F}_{\mathbb{C}}w = {\lambda w}}\right\} \) and \( {V}_{\lambda } = \{ v \in V \mid {Fv} = {\lambda v}\} \) for the \( \lambda \) -eigenspaces of \( {F}_{\mathbb{C}} \) and \( F \) in \( {V}_{\mathbb{C}} \) and \( V \) respectively. By Corollary 19.2,... | Yes |
Proposition 19.3 Every anti-self-adjoint operator \( F \) on a real Euclidean space \( V \) can be written in an appropriate orthonormal basis of \( V \) as a block diagonal matrix\n\n\[ \left( \begin{matrix} {A}_{1} & & 0 & \\ & {A}_{2} & & \\ & & \ddots & \\ 0 & & & {A}_{k} \end{matrix}\right) \text{, where }{A}_{k} ... | Proof In the notation introduced during the proof of Proposition 19.2, we again have an orthogonal decomposition (19.14), but now all eigenvalues \( \lambda \in \operatorname{Spec}{F}_{\mathbb{C}} \) are pure imaginary. Since the characteristic polynomial \( {\chi }_{{F}_{\mathbb{C}}} = {\chi }_{F} \) has real coeffici... | Yes |
Lemma 19.2 Let \( F : U \rightarrow W \) be an arbitrary \( \mathbb{C} \) -linear operator between Hermitian spaces. Then both operators \( F{F}^{ \dagger } \in \operatorname{End}\left( W\right) ,{F}^{ \dagger }F \in \operatorname{End}\left( U\right) \) are selfadjoint and have nonnegative spectra. If the operator \( F... | Proof The self-adjointness of both operators is obvious. By Corollary 19.2 on p. 491, it forces their eigenvalues to be real. If \( F{F}^{ \dagger }w = {\lambda w} \neq 0 \) for some \( w \in W \) , then \( {F}^{ \dagger }w \neq 0 \) and \( \lambda \cdot \left( {w, w}\right) = \left( {{\lambda w}, w}\right) = \left( {F... | Yes |
Theorem 19.2 (Polar Decomposition) Every invertible \( \\mathbb{C} \) -linear operator \( F \) on a finite-dimensional Hermitian space admits unique factorizations \( F = {S}_{1}{I}_{1} \) and \( F = {I}_{2}{S}_{2} \), where the operators \( {U}_{1},{U}_{2} \) are unitary, and the operators \( {S}_{1},{S}_{2} \) are se... | Proof Choose bases such that \( F{F}^{ \\dagger } \) and \( {F}^{ \\dagger }F \) are diagonal \( {}^{22} \) and put \( {S}_{1} = \\sqrt{F{F}^{ \\dagger }} \) , \( {S}_{2} = \\sqrt{{F}^{ \\dagger }F} \) as the diagonal operators obtained by extraction of the positive square roots from the diagonal elements of the corres... | Yes |
Theorem 19.3 For every \( \mathbb{C} \) -linear map of Hermitian spaces \( F : U \rightarrow W \), there exist orthonormal bases of \( U, W \) such that the matrix of \( F \) in these bases is diagonal with real nonnegative diagonal elements. Up to permutations, the diagonal elements do not depend on the choice of such... | Proof Since the endomorphism \( {F}^{ \dagger }F : U \rightarrow U \) is self-adjoint, there exists an orthonormal basis \( {e}_{1},{e}_{2},\ldots ,{e}_{n} \) in \( U \) formed by the eigenvectors of \( {F}^{ \dagger }F \) . By Lemma 19.2, the spectrum of \( {F}^{ \dagger }F \) is real and nonnegative. Therefore, \( {F... | No |
Example 19.3 (Euclidean Angles Between Subspaces) Let \( U, W \) be two vector subspaces in a real Euclidean vector space, \( \dim U = n \leq m = \dim W \) . Write \( \pi : U \rightarrow W \) for the projection along \( {W}^{ \bot } \) and \( {\alpha }_{1} \geq {\alpha }_{2} \geq \cdots \geq {\alpha }_{n} \) for the si... | \[ 0 \leq {\varphi }_{1} \leq {\varphi }_{2} \leq \cdots \leq {\varphi }_{n} \leq \pi /2,\;{\varphi }_{i} = \measuredangle \left( {{w}_{i},{u}_{i}}\right) ,\] (19.18) between the first \( n \) vectors of some orthonormal basis \( {w}_{1},{w}_{2},\ldots ,{w}_{m} \) of \( W \) and vectors \( {u}_{1},{u}_{2},\ldots ,{u}_{... | Yes |
Lemma 20.1 Take \( \mathbf{i},\mathbf{j},\mathbf{k} \) from (20.7) as the standard orienting basis in the Euclidean space of pure imaginary quaternions \( I \simeq {\mathbb{R}}^{3} \) . Then \( \operatorname{Im}\left( {pq}\right) = p \times q \) for all \( p, q \in I \) . In particular, \( {pq} \) is real if and only i... | Proof Since both maps \( p, q \mapsto p \times q \) and \( p, q \mapsto \operatorname{Im}\left( {pq}\right) \) are bilinear, it is enough to check the relation \( \operatorname{Im}\left( {pq}\right) = p \times q \) for nine pairs of basic vectors \( p, q = \mathbf{i},\mathbf{j},\mathbf{k} \) . This is exactly the multi... | Yes |
Lemma 20.2 Two arbitrary quaternions \( p, q \in \mathbb{H} \) are orthogonal if and only if \( p{q}^{ * } \in I \) . Two pure imaginary quaternions \( p, q \in I \) are orthogonal if and only if \( {pq} = - {qp} \), and in this case, \( {pq} = - {qp} \in I \) is perpendicular to the plane spanned by \( p, q \) . | Proof The first statement follows directly from formula (20.10). All other claims follow from Lemma 20.1. | No |
Lemma 20.3 The solution set of the equation \( {x}^{2} = - 1 \) in \( \mathbb{H} \) is the unit sphere \( {S}^{2} = \{ x \in I \mid \parallel x\parallel = 1\} \) in \( I \simeq {\mathbb{R}}^{3}. \) | Proof The equation \( {x}^{2} = - 1 \) forces \( \parallel x\parallel = 1 \) . Then \( {x}^{ * } = {x}^{-1} = - x \), and therefore \( x \in I \) . Conversely, for every \( x \in {S}^{2} \), we have \( {x}^{2} = - {x}^{ * }x = - {x}^{-1}x = - 1 \) . | Yes |
Lemma 20.4 Three arbitrary quaternions \( \mathbf{i},\mathbf{j},\mathbf{k} \) satisfy the relations (20.8) if and only if they form a positive orthonormal basis in \( I \), where the orientation in \( I \) is given by the initial orthonormal basis (20.7). | Proof The relations \( {i}^{2} = {j}^{2} = {k}^{2} = - 1 \) mean that all three quaternions lie on the unit sphere \( {S}^{2} \subset I \) . Then by Lemma 20.2, the relations \( \mathbf{i} \cdot \mathbf{j} = \mathbf{k} = - \mathbf{j} \cdot \mathbf{i} \) mean that \( \mathbf{k} \) is perpendicular to both \( \mathbf{i} ... | Yes |
Lemma 20.5 The operator \( {\left. {F}_{\psi }\right| }_{I} \in {\mathrm{{SO}}}_{\mathrm{{det}}}\left( I\right) \) is rotation about the line \( {\ell }_{\psi } \) by the angle \( 2\operatorname{Arg}\left( \psi \right) \) viewed in the direction of the vector \( \mathbf{l} \in {\ell }_{\psi } \) . | Proof Fit the vector \( \mathbf{l} \) into the positively oriented orthonormal basis \( \mathbf{l},\mathbf{m},\mathbf{n} \) in \( I \) . By Lemma 20.4, the multiplication table of quaternions \( l,\mathbf{m},\mathbf{n} \) is \( {\mathbf{l}}^{2} = {\mathbf{m}}^{2} = {\mathbf{n}}^{2} = \) \( \mathbf{{lmn}} = - 1 \) as in... | Yes |
Lemma 1.1 For every two universal multilinear maps\n\n\\[ \n{\\tau }_{1} : {V}_{1} \\times {V}_{2} \\times \\cdots \\times {V}_{n} \\rightarrow {U}_{1},{\\tau }_{2} : {V}_{1} \\times {V}_{2} \\times \\cdots \\times {V}_{n} \\rightarrow {U}_{2} \n\\]\n\n there exists a unique linear isomorphism \\( \\iota : {U}_{1} \\si... | Proof By the universal properties of \\( {\\tau }_{1},{\\tau }_{2} \\), there exists a unique pair of linear maps\n\n\\[ \n{F}_{21} : {U}_{1} \\rightarrow {U}_{2}\\;\\text{ and }\\;{F}_{12} : {U}_{2} \\rightarrow {U}_{1} \n\\]\n\n that fit in the commutative diagram\n\n \\mapsto \\left\\lbrack {{v}_{1}\\ldots {v}_{n}}\\right\\rbrack \\left( {\\;\\operatorname{mod}\\;\\mathcal{R}}\\right) ,\n\\]\n\nis the universal multilinear map. | Proof The multilinearity of \\( \\tau \\) is expressed exactly by the relations (1.13), which hold by definition. Let us check the universal property. For every map of sets\n\n\\[ \n\\varphi : {V}_{1} \\times {V}_{2} \\times \\cdots \\times {V}_{n} \\rightarrow W,\n\\]\n\nthere exists a unique linear map \\( F : \\math... | Yes |
Theorem 1.1 (Tensor Product of Free Modules) Let modules \( {V}_{i} \) be free with a (not necessarily finite) basis \( {E}_{i} \) . Then the tensor product \( {V}_{1} \otimes {V}_{2} \otimes \cdots \otimes {V}_{n} \) is free with a basis formed by the tensor products of basis vectors\n\n\[ {e}_{1} \otimes {e}_{2} \oti... | Proof Let us temporarily consider the symbols (1.14) just as formal records, and write \( \mathcal{W} \) for the free module with a basis formed by all these records. By Sect. 1.1.1, there exists a unique multilinear map \( \tau : {V}_{1} \times {V}_{2} \times \cdots \times {V}_{n} \rightarrow \mathcal{W} \) such that ... | Yes |
Proposition 1.1 (Commutativity Isomorphism) The map\n\n\[ \nU \otimes W \simeq W \otimes U, u \otimes w \mapsto w \otimes u, \n\]\n\nis a well-defined linear isomorphism. | Proof Since the prescription \( u \otimes w \mapsto w \otimes u \) is bilinear in \( u, w \), it assigns the well-defined homomorphism of \( K \) -modules \( U \otimes W \rightarrow W \otimes U \) . For the same reason, there exists the well-defined \( K \) -linear map \( W \otimes U \rightarrow U \otimes W, w \otimes ... | Yes |
Proposition 1.2 (Associativity Isomorphism) The maps\n\n\[ \nV \otimes \left( {U \otimes W}\right) \Leftarrow V \otimes U \otimes W ⤳ \left( {V \otimes U}\right) \otimes W \]\n\n taking \( v \otimes u \otimes w \), respectively, to \( v \otimes \left( {u \otimes w}\right) \) and \( \left( {v \otimes u}\right) \otimes w... | Proof The tensor \( v \otimes \left( {u \otimes w}\right) \in V \otimes \left( {U \otimes W}\right) \) depends 3-linearly on \( \left( {v, u, w}\right) \) . Hence, by Lemma 1.3, there exists the well-defined linear map \( V \otimes U \otimes W \rightarrow V \otimes \left( {U \otimes W}\right) \) , \( v \otimes u \otime... | Yes |
Proposition 1.3 (Distributivity Isomorphisms) For every K-module \( V \) and family of \( K \) -modules \( {U}_{x}, x \in X \), the maps\n\n\[ V \otimes \left( {{\bigoplus }_{x \in X}{U}_{x}}\right) \simeq {\bigoplus }_{x \in X}\left( {V \otimes {U}_{x}}\right) ,\;v \otimes {\left( {u}_{x}\right) }_{x \in X} \mapsto {\... | Proof It is enough to prove only (1.22). Then (1.23) follows by the commutativity isomorphism from Proposition 1.1. The map (1.22) is well defined, because the family \( {\left( v \otimes {u}_{x}\right) }_{x \in X} \) depends bilinearly on the vector \( v \in V \) and the family\n\n\[ {\left( {u}_{x}\right) }_{x \in X}... | Yes |
Example 1.4 (Kronecker Matrix Product) Consider two vector spaces \( U, W \) with bases \( {u}_{1},{u}_{2},\ldots ,{u}_{n} \) and \( {w}_{1},{w}_{2},\ldots ,{w}_{m} \) respectively, and let the linear operators \( f : U \rightarrow U, g : W \rightarrow W \) have matrices \( F = \left( {\varphi }_{ij}\right) \) and \( G... | \[ f \otimes g\left( {{u}_{j} \otimes {w}_{\ell }}\right) = \left( {\mathop{\sum }\limits_{i}{u}_{i}{\varphi }_{ij}}\right) \otimes \left( {\mathop{\sum }\limits_{k}{w}_{k}{\gamma }_{k\ell }}\right) = \mathop{\sum }\limits_{{i, k}}{\varphi }_{ij}{\gamma }_{k\ell } \cdot {u}_{i} \otimes {w}_{k}. \] This matrix is called... | Yes |
Lemma 1.4 For every epimorphism of \( K \) -modules \( f : U \rightarrow W \) and every \( K \) -module \( V \), the map \( {\operatorname{Id}}_{V} \otimes f : V \otimes U \rightarrow V \otimes W \) is surjective. | Proof All decomposable tensors \( v \otimes w \in V \otimes W \) certainly lie in the image of \( f \otimes {\operatorname{Id}}_{V} \) . | No |
Lemma 1.5 For every monomorphism of \( K \) -modules \( f : U \hookrightarrow W \) and every free \( K \) -module \( F \), the map \( {\operatorname{Id}}_{F} \otimes f : F \otimes U \rightarrow F \otimes W \) is injective. | Proof If \( F \simeq K \) has rank one, then the multiplication maps\n\n\[ K \otimes U \simeq U,\;\lambda \otimes u \mapsto {\lambda u}, \]\n\n\[ K \otimes W \simeq W,\;\mu \otimes w \mapsto {\mu w}, \]\n\nare bijective by Exercise 1.5, and they transform the map \( {\operatorname{Id}}_{F} \otimes f : K \otimes U \righ... | No |
Proposition 2.1 (Universal Property of Free Associative Algebras) For every associative \( \mathbb{k} \) -algebra \( A \) with unit and \( \mathbb{k} \) -linear map \( f : V \rightarrow A \), there exists a unique homomorphism of \( k \) -algebras \( \widetilde{f} : \mathrm{T}V \rightarrow A \) such that \( f = \wideti... | Proof (of Proposition 2.1) A homomorphism of \( \mathbb{k} \) -algebras \( \widetilde{f} : \top V \rightarrow A \) such that \( f = \widetilde{f} \circ \iota \) maps every decomposable tensor \( {v}_{1} \otimes {v}_{2} \otimes \cdots \otimes {v}_{n} \) to the product \( f\left( {v}_{1}\right) \cdot f\left( {v}_{2}\righ... | Yes |
Proposition 2.2 For a finite-dimensional vector space \( V \), the pairing (2.3) is perfect, i.e., the linear map (2.2) is an isomorphism. | Proof Choose dual bases \( {e}_{1},{e}_{2},\ldots ,{e}_{n} \in V \) and \( {x}_{1},{x}_{2},\ldots ,{x}_{n} \in {V}^{ * } \) . Then the tensor monomials \( {e}_{{i}_{1}} \otimes {e}_{{i}_{2}} \otimes \cdots \otimes {e}_{{i}_{r}} \) and \( {x}_{{j}_{1}} \otimes {x}_{{j}_{2}} \otimes \cdots \otimes {x}_{{j}_{s}} \) form b... | No |
For every finite-dimensional vector space \( V \), there is a canonical isomorphism\n\n\[ \n{\left( {V}^{ * }\right) }^{\otimes n} \simeq \operatorname{Hom}\left( {V,\ldots, V;k}\right) \n\]\n\nmapping the decomposable tensor \( \vartheta = {\xi }_{1} \otimes {\xi }_{2} \otimes \cdots \otimes {\xi }_{n} \in {V}^{* \oti... | Proof The universal property of tensor product \( {V}^{\otimes n} \) asserts that the dual space \( {\left( {V}^{\otimes n}\right) }^{ * } \), that is, the space of linear maps \( {V}^{\otimes n} \rightarrow \mathbb{k} \), is isomorphic to the space of \( n \) -linear forms \( V \times V \times \cdots \times V \rightar... | No |
Consider an \( n \) -linear form \( \varphi : V \times V \times \cdots \times V \rightarrow \mathbb{k} \) as a tensor from \( {V}^{* \otimes n} \) by means of the isomorphism from Corollary 2.1, and contract this tensor with a vector \( v \in V \) at the first tensor factor. The result of such a contraction is called t... | \[ v \sqsubset \varphi \left( {{u}_{1},{u}_{2},\ldots ,{u}_{n - 1}}\right) = \varphi \left( {v,{u}_{1},{u}_{2},\ldots ,{u}_{n - 1}}\right) \] for all \( {u}_{1},{u}_{2},\ldots ,{u}_{n - 1} \in V \) . Indeed, since both sides of the equality are linear in \( \varphi \) , it is enough to verify it only for the \( n \) -l... | Yes |
Theorem 2.1 For every \( t \in {V}^{\otimes n} \), the subspace \( \operatorname{Supp}\left( t\right) \subset V \) is spanned by the images of the \( n \) ! contraction maps (2.7) corresponding to all possible choices of \( J \) . | Proof Let \( \operatorname{Supp}\left( t\right) = W \subset V \) . We have to show that every linear form \( \xi \in {V}^{ * } \) annihilating all the subspaces \( \operatorname{im}\left( {c}_{t}^{I}\right) \subset W \) has to annihilate all of \( W \) as well. Assume the contrary. Let \( \xi \in {V}^{ * } \) be a line... | Yes |
Proposition 2.3 The universal symmetric n-linear map\n\n\[ \n{\sigma }_{n} : V \times V \times \cdots \times V \rightarrow U \]\n\nis provided by tensor multiplication followed by factorization through the commutativity relations, i.e.,\n\n\[ \n{\sigma }_{n} : V \times V \times \cdots \times V - \overset{\tau }{ - } \s... | Proof By the universal property of tensor multiplication \( \tau : V \times V \times \cdots \times V \rightarrow {V}^{\otimes n} \) , every \( n \) -linear map \( \varphi : V \times V \times \cdots \times V \rightarrow W \) is uniquely factorized as \( \varphi = \widetilde{F} \circ \tau \) for some linear map \( \widet... | Yes |
For an arbitrary (not necessarily finite-dimensional) vector space \( V \), the nth symmetric power \( {S}^{n}V \) and the space \( {\operatorname{Sym}}^{n}\left( {V,\mathbb{k}}\right) \) of symmetric \( n \) -linear forms \( V \times V \times \cdots \times V \rightarrow \mathbb{k} \) are canonically dual to each other... | Proof Right composition with the commutative multiplication\n\n\[{\sigma }_{n} : V \times V \times \cdots \times V \rightarrow {S}^{n}V\]\n\nwhich takes a covector \( \xi : {S}^{n}V \rightarrow \mathbb{k} \) to the symmetric \( n \) -linear form\n\n\[ \xi \circ {\sigma }_{n} : V \times V \times \cdots \times V \rightar... | Yes |
Corollary 2.3 For an arbitrary (not necessarily finite-dimensional) vector space \( V \) , the nth exterior power \( {\Lambda }^{n}V \) and the space \( {\operatorname{Alt}}^{n}\left( {V,\mathbb{k}}\right) \) of alternating \( n \) -linear forms \( V \times V \times \cdots \times V \rightarrow \mathbb{k} \) are canonic... | Proof The same as for Corollary 2.2 on p. 28. | No |
For every basis \( {e}_{1},{e}_{2},\ldots ,{e}_{d} \) of \( V \), a basis in \( {\Lambda }^{d}V \) is formed by the Grassmannian monomials\n\n\[ \n{e}_{I}\overset{\text{ def }}{ = }{e}_{{i}_{1}} \land {e}_{{i}_{2}} \land \cdots \land {e}_{{i}_{n}} \]\n\n(2.12)\n\nnumbered by all \( I = \left( {{i}_{1},{i}_{2},\ldots ,{... | Proof Write \( U \) for the vector space of dimension \( \left( \begin{array}{l} d \\ n \end{array}\right) \) with the basis \( \left\{ {u}_{I}\right\} \) numbered by the same multi-indices \( I \) as the Grassmannian monomials (2.12). We know from Sect. 1.1.1 on p. 1 that every \( n \) -linear map \( \alpha : V \times... | Yes |
Example 2.2 (Tensor Square Decomposition) For \( n = 2 \), the symmetrization and alternation maps form a pair of complementary projectors, \( {}^{4} \) that is, | \[ {\mathrm{{sym}}}_{2} + {\mathrm{{alt}}}_{2} = \left( {\mathrm{{Id}} + {s}_{12}}\right) /2 + \left( {\mathrm{{Id}} - {s}_{12}}\right) /2 = \mathrm{{Id}}, \] where \( {s}_{12} \in {S}_{2} \) is a transposition. Therefore, there exists the direct sum decomposition \[ {V}^{\otimes 2} = {\operatorname{Sym}}^{2}V \oplus {... | Yes |
Proposition 2.5 If \( \operatorname{char}\left( k\right) = 0 \), then the restriction of the quotient map\n\n\[ \n{V}^{\otimes n} \rightarrow {S}^{n}V\n\]\n\nto the subspace \( {\operatorname{Sym}}^{n} \subset {V}^{\otimes n} \) and the restriction of the quotient map\n\n\[ \n{V}^{\otimes n} \rightarrow {\Lambda }^{n}V... | Proof The projection \( {\pi }_{\mathrm{{sym}}} \) maps each of the \( n!/\left( {{m}_{1}!{m}_{2}!\cdots {m}_{d}!}\right) \) summands in (2.21) to the commutative monomial \( {e}_{1}^{{m}_{1}}{e}_{2}^{{m}_{2}}\ldots {e}_{d}^{{m}_{d}} \) . Similarly, the projection \( {\pi }_{\mathrm{{sk}}} \) sends each of the \( n \) ... | Yes |
Proposition 2.6 Let \( V \) be a vector space, not necessarily finite-dimensional, over a field \( k \) of characteristic zero. Then for every homogeneous polynomial \( f \in {S}^{n}{V}^{ * } \) , | Proof Consider the expansion (2.28) from Exercise 2.15 for \( k = n = \deg f \) . Its right-hand side contains the unique term depending on all the vectors \( {v}_{1},{v}_{2},\ldots ,{v}_{n} \) , namely \( n! \cdot \widetilde{f}\left( {{v}_{1},{v}_{2},\ldots ,{v}_{n}}\right) \) . For every proper subset \( I \subsetneq... | No |
Example 2.4 (Taylor Expansion) For \( k = 2 \), the expansion (2.28) from Exercise 2.15 turns into the identity\n\n\[ f\left( {u + w}\right) = \widetilde{f}\left( {u + w, u + w,\ldots, u + w}\right) = \mathop{\sum }\limits_{{m = 0}}^{n}\left( \begin{matrix} n \\ m \end{matrix}\right) \cdot \widetilde{f}\left( {{u}^{m},... | where \( n = \deg f \), which holds for every polynomial \( f \in {S}^{n}{V}^{ * } \) and all vectors \( u, w \in V \) . The relations (2.34) allow us to rewrite this identity as the Taylor expansion for \( f \) at \( u \) :\n\n\[ f\left( {u + w}\right) = \mathop{\sum }\limits_{{m = 0}}^{{\deg f}}\frac{1}{m!}{\partial ... | Yes |
Proposition 2.7 Let \( \mathbb{k} \) be a field of characteristic zero, \( V \) a finite-dimensional vector space over \( \mathbb{k} \), and \( f \in {S}^{n}{V}^{ * } \) a polynomial written in the form (2.39) in some basis of \( {V}^{ * } \) . If \( f = {\varphi }^{n} \) is the proper nth power of some linear form \( ... | Proof The equality \( f = {\varphi }^{n} \) means that \( \operatorname{Supp}\left( f\right) \subset {V}^{ * } \) is the 1-dimensional subspace spanned by \( \varphi \) . In this case, all linear forms (2.40) are proportional to \( \varphi \) . Such a form \( \psi = {\lambda \varphi } \) has \( {\psi }^{n} = f \) if an... | Yes |
Example 2.5 (Binary Forms of Rank 1) We know from Example 11.6 of Algebra I that a homogeneous binary form of degree \( n \) ,\n\n\[ f\left( {{x}_{0},{x}_{1}}\right) = \mathop{\sum }\limits_{k}{a}_{k} \cdot \left( \begin{array}{l} n \\ k \end{array}\right) \cdot {x}_{0}^{n - k}{x}_{1}^{k}, \]\n\nis the proper \( n \) t... | This is equivalent to the condition\n\n\[ \operatorname{rk}\left( \begin{array}{llll} {a}_{0} & {a}_{1} & \ldots & {a}_{n - 1} \\ {a}_{1} & {a}_{2} & \ldots & {a}_{n} \end{array}\right) = 1 \]\n\nwhich is expanded to a system of homogeneous quadratic equations \( {a}_{i}{a}_{j + 1} = {a}_{i + 1}{a}_{j} \) in the coeffi... | Yes |
Proposition 2.8 The following conditions on a Grassmannian polynomial \( \omega \in {\Lambda }^{n}V \) written in the form (2.47) are equivalent:\n\n1. \( \omega = {u}_{1} \land {u}_{2} \land \cdots \land {u}_{n} \) for some \( {u}_{1},{u}_{2},\ldots ,{u}_{n} \in V \) .\n\n2. \( u \land \omega = 0 \) for all \( u \in \... | Proof Condition 1 holds if and only if \( \omega \) belongs to the top homogeneous component of its linear span, \( \omega \in {\Lambda }^{\dim \operatorname{Supp}\left( \omega \right) }\operatorname{Supp}\left( \omega \right) \) . Condition 2 means the same because of the following exercise.\n\nExercise 2.22 Show that... | No |
Example 2.6 (The Plücker Quadric) Let \( n = 2,\dim V = 4 \), and let \( {e}_{1},{e}_{2},{e}_{3},{e}_{4} \) be a basis of \( V \) . Then the expansion (2.47) for \( \omega \in {\Lambda }^{2}V \) looks like \( \omega = \mathop{\sum }\limits_{{i, j}}{a}_{ij}{e}_{i} \land {e}_{j} \) , where the coefficients \( {a}_{ij} \)... | \[ {a}_{12}{a}_{34} - {a}_{13}{a}_{24} + {a}_{14}{a}_{23} = 0. \] (2.50) All other choices of \( \left( {{i}_{1},{i}_{2},{i}_{3}}\right) \) and \( {j}_{1} \notin \left\{ {{i}_{1},{i}_{2},{i}_{3}}\right\} \) lead to exactly the same relation. | Yes |
Example 2.7 (The Plücker Quadric, Geometric Continuation of Example 2.6) For \( \dim V = 4 \), the Grassmannian \( \operatorname{Gr}\left( {2,4}\right) = \operatorname{Gr}\left( {2, V}\right) \) can be viewed as the set of lines \( \ell = \mathbb{P}\left( U\right) \) in \( {\mathbb{P}}_{3} = \mathbb{P}\left( V\right) \... | \[ P = \left\{ {\omega \in {\Lambda }^{2}V \mid \omega \land \omega = 0}\right\} \] in \( {\mathbb{P}}_{5} \), called the Plücker quadric. | Yes |
Problem 2.4 (Aronhold’s Principle) Let \( V \) be a finite-dimensional vector space over a field \( \mathbb{k} \) of zero characteristic. Prove that the subspace of symmetric tensors \( {\operatorname{Sym}}^{n}\left( V\right) \subset {V}^{\otimes n} \) is linearly generated by the proper \( n \) th tensor powers \( {v}... | Write the symmetric tensor\n\n\[ u \otimes w \otimes w + w \otimes u \otimes w + w \otimes w \otimes u \in {\operatorname{Sym}}^{3}\left( V\right) \]\n\n as a linear combination of proper tensor cubes. | No |
Verify that the Taylor expansion for the polynomial \( \det \left( A\right) \) in the space of linear operators \( A : V \rightarrow V \) has the following form: | \[ \det \left( {{\lambda A} + {\mu B}}\right) = \mathop{\sum }\limits_{{p + q = n}}{\lambda }^{p}{\mu }^{q} \cdot \operatorname{tr}\left( {{\Lambda }^{p}A \cdot {\Lambda }^{q}{B}^{ * }}\right) ,\] | No |
Prove that the answers you got in the previous two problems hold for nondiagonalizable linear operators \( F \) as well. Use the following arguments, known as a splitting principle. Interpret the relation on \( F \) you are going to prove as the identical vanishing of some polynomial with rational coefficients in the m... | (a) If a polynomial \( f \in \mathbb{Q}\left\lbrack {{x}_{1},{x}_{2},\ldots ,{x}_{n}}\right\rbrack \) evaluates to zero at all points of some dense subset of \( {\mathbb{C}}^{n} \), then \( f \) is the zero polynomial. (Thus, it is enough to check that the relation being proved holds for some set of complex matrices de... | Yes |
Problem 2.23 (Grassmannian Exponential) Let \( V \) be a vector space over a field \( \mathbb{k} \) of arbitrary characteristic. The Grassmannian exponential is defined for decomposable \( \omega \in {\Lambda }^{2m} \) by the assignment \( {e}^{\omega }\overset{\text{ def }}{ = }1 + \omega \) . For an arbitrary even-de... | Prove that the exponential map \( {\Lambda }^{\text{even }}V \hookrightarrow {\Lambda }^{\text{even }}V,\zeta \mapsto {e}^{\zeta } \), is an injective homomorphism of the additive group of even-degree Grassmannian polynomials to the multiplicative group of even-degree Grassmannian polynomials with unit constant term. S... | No |
Prove that neither operator depends on the choice of basis in \( V \) and that both operators have zero squares, \( {d}^{2} = 0 = {\partial }^{2} \) . Verify that their \( s \) -commutator \( d\partial + \partial d \) acts on \( {\Lambda }^{k}V \otimes {S}^{m}V \) as a homothety \( \left( {k + m}\right) \cdot \mathrm{{... | \[ d\overset{\text{ def }}{ = }\mathop{\sum }\limits_{v}{\xi }_{v} \otimes \frac{\partial }{\partial {x}_{v}} : {\Lambda }^{k}V \otimes {S}^{m}V \rightarrow {\Lambda }^{k + 1}V \otimes {S}^{m - 1}V, \] \[ \partial \triangleq \mathop{\sum }\limits_{\nu }\frac{\partial }{\partial {\xi }_{\nu }} \otimes {x}_{\nu } : {\Lam... | No |
Proposition 3.2 The polynomials \( {e}_{\lambda } = {e}_{{\lambda }_{1}}{e}_{{\lambda }_{2}}\cdots {e}_{{\lambda }_{m}} \), where \( \lambda \) runs through the Young diagrams with at most \( n \) columns, form a basis of the \( \mathbb{Z} \) -module of symmetric polynomials in \( n \) variables. | Proof Write the basis vectors \( {m}_{\mu } \) in the lexicographically increasing order of their indices \( \mu \), and the polynomials \( {e}_{\lambda } \) in the lexicographically increasing order of the transposed diagrams \( {\lambda }^{t} \) . Then the transition matrix from \( {e}_{\lambda } \) to \( {m}_{\mu } ... | Yes |
Proposition 3.3 There exists a unique involutive automorphism \( \omega \) of the ring of symmetric polynomials in \( n \) variables such that \( \omega \left( {e}_{k}\right) = {h}_{k} \) and \( \omega \left( {h}_{k}\right) = {e}_{k} \) for every \( k = 1,2,\ldots, n \) . | Proof Since the ring of symmetric polynomials is \( \mathbb{Z}\left\lbrack {{e}_{1},{e}_{2},\ldots ,{e}_{n}}\right\rbrack \), the assignment \( \omega : {e}_{k} \mapsto {h}_{k} \) is uniquely extended to a ring endomorphism of the ring of symmetric polynomials. The recurrence formulas (3.10),(3.11) show that \( \omega ... | Yes |
The symmetric Newton polynomials \( {p}_{1},{p}_{2},\ldots ,{p}_{n} \) are algebraically independent. Every symmetric polynomial in \( \mathbb{Q}\left\lbrack {{x}_{1},{x}_{2},\ldots ,{x}_{n}}\right\rbrack \) can be uniquely written as a polynomial with rational coefficients in \( {p}_{1},{p}_{2},\ldots ,{p}_{n} \) . In... | Proof The formula (3.14) implies that for every \( N \in \mathbb{N} \), the \( \mathbb{Q} \) -linear span of products \( {p}_{1}^{{m}_{1}}{p}_{2}^{{m}_{2}}\cdots {p}_{n}^{{m}_{n}} \) in the vector space \( \mathbb{Q}{\left\lbrack {x}_{1},{x}_{2},\ldots ,{x}_{n}\right\rbrack }_{ \leq N} \) of polynomials of total degree... | Yes |
The elementary and complete symmetric polynomials \( {e}_{k},{h}_{k} \) are expanded as rational linear combinations of monomials \( {p}_{\lambda } \) by the formulas\n\n\[ \n{h}_{k} = \mathop{\sum }\limits_{{\left| \lambda \right| = k}}{z}_{\lambda }^{-1}{p}_{\lambda }\n\]\n\n(3.17)\n\n\[ \n{e}_{k} = \mathop{\sum }\li... | Proof Formulas (3.17) and (3.18) are transferred one to the other by the involution \( \omega \) . Thus, it is enough to prove only the first of them. Recall that\n\n\[ \nP\left( t\right) = \mathop{\sum }\limits_{{k \geq 1}}{p}_{k}\left( x\right) \cdot {t}^{k - 1} = \frac{d}{dt}\log H\left( t\right) .\n\]\n\nTherefore,... | Yes |
Example 3.1 For \( k = 3 \), we get the expression \( {e}_{3} = {p}_{3} - \frac{1}{2}{p}_{1}{p}_{2} + \frac{1}{6}{p}_{1}^{3} \), which agrees with the multinomial formula | \[ {\left( {x}_{1} + \cdots + {x}_{n}\right) }^{3} = \sum {x}_{i}^{3} + 3\mathop{\sum }\limits_{{i \neq j}}{x}_{i}{x}_{j}^{2} + 6\mathop{\sum }\limits_{{i < j < k}}{x}_{i}{x}_{j}{x}_{k}. \] | Yes |
For \( n = 2 \), the Giambelli formula gives the following expression for \( {s}_{\left( 2,1\right) } \) in \( \mathbb{Z}\left\lbrack {{x}_{1},{x}_{2}}\right\rbrack \) : | \[ {s}_{\left( 2,1\right) } = \det \left( \begin{matrix} {h}_{2} & {h}_{3} \\ 1 & {h}_{1} \end{matrix}\right) = {h}_{1}{h}_{2} - {h}_{3} = {e}_{1}{e}_{2} - {e}_{3}. \] | Yes |
Corollary 3.4 (Pieri’s Formula) We have \( {s}_{\lambda } \cdot {h}_{k} = \mathop{\sum }\limits_{\mu }{s}_{\mu } \), where \( \mu \) runs through the Young diagrams of length at most \( n \) obtained from \( \lambda \) by adding \( k \) cells in \( k \) different columns. | Proof Let \( v = \lambda + \delta ,\eta = \mu + \delta \), where \( \delta = \left( {n - 1, n - 2,\ldots ,1,0}\right) \) and \( \lambda ,\mu \) are Young diagrams with lengths of rows \( {\lambda }_{i} = {v}_{i} - n + i,{\mu }_{i} = {\eta }_{i} - n + i \) . In terms of \( \lambda ,\mu \) , the inequalities \( {\eta }_{... | Yes |
Problem 3.17 (The Second Giambelli Formula) Prove that\n\n\[ \n{s}_{{\lambda }^{t}} = \det \left( \begin{matrix} {e}_{{\lambda }_{1}} & {e}_{{\lambda }_{1} + 1} & \ldots & {e}_{{\lambda }_{1} + n - 1} \\ & & & \\ {e}_{{\lambda }_{2} - 1} & {e}_{{\lambda }_{2}} & \ddots & \vdots \\ \vdots & \ddots & \ddots & {e}_{{\lamb... | \[ {}^{15} \] See Proposition 9.4 from Algebra I. | No |
Lemma 4.1 Every horizontal operation preserves stable matchings between rows, meaning that all free balls remain free and all coupled pairs of balls remain coupled in the same pairs after the operation is applied. Similarly, every vertical operation preserves stable matchings between columns. | Proof Let us fix a stable matching between the \( \left( {j + 1}\right) \) th and \( j \) th rows in an array \( a \), and verify that all operations \( {L}_{i} \) preserve this matching. It is clear when \( {L}_{i} \) does nothing with \( a \) . Let \( {L}_{i} \) move a ball \( \beta \) . If \( \beta \) lies neither i... | Yes |
Corollary 4.1 Every horizontal operation \( {L}_{i},{R}_{i} \) commutes with every vertical operation \( {D}_{j},{U}_{j} \) . | Proof Let us show, for example, that \( {D}_{j}{L}_{i} = {L}_{i}{D}_{j} \) (all other cases are completely similar). Given an array \( a \), it follows from Lemma 4.1 that the operation \( {L}_{i} \) either leaves both arrays \( a,{D}_{j}a \) unchanged or moves the same ball in \( a \) and in \( {D}_{j}a \) to the left... | Yes |
Corollary 4.2 Let \( H \) be a word built from horizontal operations \( {L}_{i},{R}_{i} \) . Then \( H \) acts effectively on an array a if and only if \( H \) acts effectively on all arrays obtained from a by means of vertical operations. Similarly, a word \( V \) built from vertical operations acts effectively on a i... | Proof The second statement is obtained from the first by means of transposition. To verify the first, it is enough to check that for every array \( a \) and all \( i, j \), the operation \( {L}_{i} \) acts effectively on \( a \) if and only if it acts effectively on \( {D}_{j}a \) and \( {U}_{j}a \) . This holds, becau... | Yes |
Proposition 4.1 For every array \( a \), the result of downward condensation of a does not depend on the choice of condensing word. The same holds for left, right, and upward condensing as well. | Proof If \( a \) is L-dense, then every D-condensing of \( a \) preserves the column weight \( {w}_{I}\left( a\right) \) and therefore leads to the bidense array corresponding to the Young diagram \( \lambda = {w}_{I}\left( a\right) \) . For an arbitrary array \( a \), let \( L = {L}_{{i}_{1}}{L}_{{i}_{2}}\ldots {L}_{{... | Yes |
Theorem 4.1 Let \( \mathcal{A},\mathcal{L},\mathcal{D},\mathcal{B} \) denote the sets of all \( m \times n \) arrays and all \( L \) -dense, D-dense, and bidense arrays respectively. The diagram in which the maps \( L, D \) send an array to its left and down condensations, is a Cartesian square. | Proof The maps \( L, D \) are well defined by Proposition 4.1 and commute by Corollary 4.1. We have to show that for every \( b \in \mathcal{B} \), the map \[ \mathcal{A} \rightarrow \mathcal{L}\underset{\mathcal{B}}{ \times }\mathcal{D},\;a \mapsto \left( {{La},{Da}}\right) \] establishes a bijection between the array... | Yes |
The graph of a map \( a : I \rightarrow J \) can be viewed as an array with exactly one ball in every column. Theorem 4.1 bijectively parameterizes such arrays by the pairs \( \left( {{a}_{\ell },{a}_{d}}\right) \), where \( {a}_{\ell } \) is L-dense, \( {a}_{d} \) is D-dense, \( D{a}_{\ell } = L{a}_{d} \), and \( {w}_... | By Sect. 4.2.3, every such a pair determines and is uniquely determined by the following data:\n\n- the shape \( \lambda \left( a\right) = \lambda \left( {a}_{\ell }\right) = \lambda \left( {a}_{d}\right) = {DLa} \in \mathcal{B} \), which is an arbitrary Young diagram \( \lambda \) of weight \( \left| \lambda \right| =... | Yes |
Example 4.2 (RSK-Type Correspondence) For \( J = I \), the construction from the previous example establishes a one-to-one correspondence between the symmetric group \( {S}_{n} \), formed by the \( n \) ! bijections \( I \simeq I \), and the pairs of standard Young tableaux of weight \( n \) . Hence,\n\n\[ \mathop{\sum... | Since the graphs of involutive permutations \( {}^{6} \) are the self-conjugate arrays \( a = {a}^{t} \), they correspond to the pairs of equal standard tableaux. Therefore,\n\n\[ \mathop{\sum }\limits_{\lambda }{d}_{\lambda } = \left| \left\{ {\sigma \in {S}_{n} \mid {\sigma }^{2} = 1}\right\} \right| \]\n\n(4.7) | No |
Example 4.3 (Complete and Elementary Symmetric Polynomials) The standard Schur polynomial \( {s}_{\left( k\right) }\left( x\right) \), indexed by the one-row Young diagram\n\n\[ \lambda = \left( {k,0,\ldots ,0}\right) = \underset{k}{\underbrace{▱▱\cdots ▱▱}}, \] | and coincides with the complete symmetric polynomial \( {}^{14}{h}_{k}\left( {{x}_{1},{x}_{2},\ldots ,{x}_{m}}\right) \), the sum of all monomials of total degree \( k \) in the \( {x}_{i} \) . Indeed, for every content vector \( \eta \in \) \( {\mathbb{Z}}_{ \geq 0}^{m} \) of weight \( \left| \eta \right| = \sum {\eta... | Yes |
Fix two collections of independent variables \( {x}_{1},{x}_{2},\ldots ,{x}_{n},{y}_{1},{y}_{2},\ldots ,{y}_{m} \) and interpret all the balls in the \( \left( {i, j}\right) \) cell of every \( I \times J \) array \( a \) as the monomials \( {x}_{i}{y}_{j} \) . Then in the notation of Sect. 4.4, the product of the ball... | \[ \mathop{\sum }\limits_{\lambda }{s}_{\lambda }\left( x\right) \cdot {s}_{\lambda }\left( y\right) = \mathop{\prod }\limits_{{i, j}}\frac{1}{1 - {x}_{i}{y}_{j}}. \] | Yes |
Theorem 4.2 (The Littlewood-Richardson Rule) In formula (4.18), the summation is over all Young diagrams \( v \) obtained by adding \( \left| \mu \right| \) extra cells to the diagram \( \lambda \) . The coefficient \( {c}_{\lambda \mu }^{v} \) in (4.18) equals the total number of fillings of the skew diagram \( v \sma... | Proof (of Theorem 4.2) For every \( v \), we have to compute a number of those DU-orbits in \( {O}_{\lambda } \otimes {O}_{\mu } \) whose left condensation is the standard orbit \( {O}_{v} \) . Let an array \( {ab} \) belong to such an orbit. Then both arrays \( a, b \) are L-dense and have \( {w}_{I}\left( a\right) = ... | No |
Proposition 4.4 The involution \( \omega : \Lambda \simeq \Lambda \) introduced in Proposition 3.3 on p. 62 acts on the Schur basis by the rule \( \omega \left( {s}_{\lambda }\right) = {s}_{{\lambda }^{t}} \), i.e., transposes Young diagrams indexing the Schur polynomials. | Proof Since the Schur polynomials \( {s}_{\lambda } \) form a basis of the \( \mathbb{Z} \) -module of symmetric functions \( \Lambda \), the assignment \( {s}_{\lambda } \mapsto {s}_{{\lambda }^{t}} \) provides \( \Lambda \) with a \( \mathbb{Z} \) -linear involution. It follows from formulas (4.23),(4.24) that this i... | Yes |
Corollary 4.3 (Second Giambelli Formula)\n\n\[ \n{s}_{{\lambda }^{t}} = \det \left( \begin{matrix} {e}_{{\lambda }_{1}} & {e}_{{\lambda }_{1} + 1} & \ddots & {e}_{{\lambda }_{1} + n - 1} \\ {e}_{{\lambda }_{2} - 1} & {e}_{{\lambda }_{2}} & \ddots & \ddots \\ \ddots & \ddots & \ddots & {e}_{{\lambda }_{n - 1} + 1} \\ {e... | Proof Apply the involution \( \omega \) to the first Giambelli formula (3.23) on p. 66. | Yes |
Proposition 4.5 The Newton polynomials \( {p}_{\lambda } \) form an orthogonal basis of the vector space of symmetric functions with rational coefficients \( \mathbb{Q} \otimes \Lambda \), and | \[ \left\langle {{p}_{\lambda },{p}_{\lambda }}\right\rangle = {z}_{\lambda } = \mathop{\prod }\limits_{k}\left( {{m}_{k}! \cdot {k}^{{m}_{k}}}\right) . \] Proof Let us expand the geometric progressions on the right-hand side of Cauchy's \( {\text{identity}}^{22} \) in terms of Newton power sums in the variables \( x \... | Yes |
Problem 4.9 Let us cut a Young diagram \( \lambda \) whose main diagonal consists of \( k \) cells into \( {k\Gamma } \) -shaped hooks \( {}^{26}{\gamma }_{1},{\gamma }_{2},\ldots ,{\gamma }_{k} \) with corners on the main diagonal of \( \lambda \) . | Formally, \( {\gamma }_{i} = \left( {{\lambda }_{i} - i + 1,{1}^{{\lambda }_{i}^{t} - i}}\right) \) for every \( i = 1,2,\ldots, k \) . | Yes |
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