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In Sect. 11.5 on p.270, we have seen that every linear automorphism \( F \in \mathrm{{GL}}\left( V\right) \) induces a bijective transformation \( \bar{F} : \mathbb{P}\left( V\right) \rightarrow \mathbb{P}\left( V\right) \) . This gives a surjective homomorphism\n\n\[ \pi : \mathrm{{GL}}\left( V\right) \rightarrow \mat... | By Theorem 11.1 on p. 270, its kernel \( \ker \pi \simeq {\mathbb{k}}^{ * } \) consists of the scalar homotheties \( v \mapsto {\lambda v},\lambda \in {\mathbb{k}}^{ * } \) . | Yes |
Example 12.10 (Surjection \( {S}_{4} \rightarrow {S}_{3} \) ) In Example 11.9 on p. 274 we attached a complete quadrangle abcd to a quadruple of points \( a, b, c, d \in {\mathbb{P}}_{2} \) such that no three of them are collinear. It is formed by three pairs of opposite edges (ab) and (cd), (ac) and (bd), (ad) and (bc... | For example, the 3- cycle \( \left( {b, c, a, d}\right) \in {S}_{4} \) leads to the cyclic permutation \( \left( {y, z, x}\right) \) ; the transpositions \( \left( {b, a, c, d}\right) ,\left( {a, c, b, d}\right) ,\left( {c, b, a, d}\right) \) lead to the transpositions \( \left( {x, z, y}\right) ,\left( {y, x, z}\right... | Yes |
Example 12.11 (Proper Group of Cube and \( {S}_{4} \) ) The proper group of the cube \( {\mathrm{{SO}}}_{\text{cube }} \subset {\mathrm{{SO}}}_{3}\left( \mathbb{R}\right) \) acts on the four lines \( a, b, c, d \) joining opposite vertices and on the three lines \( x, y, z \) joining the centers of opposite faces (see ... | \[ {\mathrm{{SO}}}_{\text{cube }} \rightarrow {S}_{4} \] (12.9) It is bijective, because both groups are of the same order 24. Under the isomorphism (12.9), six rotations by \( \pm {90}^{ \circ } \) about lines \( x, y, z \) go to six 4-cycles of cyclic type \( ▱▱▱▱▱ \), three rotations by \( {180}^{ \circ } \) about t... | Yes |
Let \( X \) be the set of all elements of a group \( G \) and \( \operatorname{Aut}\left( X\right) \) the group of set-theoretic bijections \( X \simeq X \) knowing nothing about the group structure on \( G \) . The map \[ \lambda : G \rightarrow \operatorname{Aut}X,\;g \mapsto \left( {{\lambda }_{g} : x \mapsto {gx}}\... | because \( {\lambda }_{gh}\left( x\right) = {ghx} = {\lambda }_{g}\left( {hx}\right) = {\lambda }_{g}\left( {{\lambda }_{h}\left( x\right) }\right) = {\lambda }_{g} \circ {\lambda }_{h}\left( x\right) \). It is called the left regular action of \( G \) on itself. Since the equality \( {gh} = h \) in \( G \) implies the... | Yes |
Proposition 12.2 (Orbit Length Formula) Let \( G \) be a finite transformation group of an arbitrary set and let \( x \) be any point of this set. Then\n\n\[ \left| {Gx}\right| = \left| G\right| : \left| {{\operatorname{Stab}}_{G}\left( x\right) }\right| .\n\]\n\nIn particular, the lengths of all orbits and orders of a... | Proof The group \( G \) decomposes into the disjoint union of the fibers of the surjective orbit map (12.13). All the fibers have cardinality \( \left| {\operatorname{Stab}\left( x\right) }\right| \) . | Yes |
Proposition 12.3 The stabilizers of all points lying in the same orbit are conjugate: | Proof Take \( z = y \) in the diagram (12.15). | No |
Example 12.15 (Multinomial Coefficients Revisited) Fix an alphabet\n\n\\[ \nA = \\left\\{ {{a}_{1},{a}_{2},\\ldots ,{a}_{k}}\\right\\} \n\\]\n\nof cardinality \\( k \\) and write \\( X \\) for the set of all words of length \\( n \\) in this alphabet. Equivalently, \\( X \\) can be viewed as the set of all maps \\( w :... | The \\( {S}_{n} \\) -orbit of a given word \\( w \\in X \\) consists of the words in which every letter \\( {a}_{i} \\in A \\) appears the same number of times as in \\( w \\) . Therefore, the points of the quotient \\( X/{S}_{n} \\) are naturally marked by the sequences \\( {m}_{1},{m}_{2},\\ldots ,{m}_{k} \\), where ... | Yes |
The orbits of the adjoint action Ad : \( G \rightarrow \operatorname{Aut}\left( G\right) \) are called conjugation classes in \( G \) . Such a class \( \operatorname{Ad}\left( G\right) h = \left\{ {{gh}{g}^{-1} \mid g \in G}\right\} \) consists of all elements conjugate to a given element \( h \in G \) . Let us describ... | The adjoint orbit corresponding to the diagram \( \lambda \) consists of all permutations obtained as follows: fill the cells of \( \lambda \) by the numbers \( 1,2,\ldots, n \) without repetitions and form the product of cycles recorded in the rows of diagram. The adjoint action of an \( {}^{20} \) That is, consisting... | Yes |
Theorem 12.2 (Burnside-Pólya-Redfield Formula) Let a finite group \( G \) act on a finite set \( X \) . For each \( g \in G \), let \( {X}^{g} \) be the fixed-point set of the transformation \( g \), i.e., \( {X}^{g} = \{ x \in X \mid {gx} = x\} = \{ x \in X \mid g \in \operatorname{Stab}\left( x\right) \} \) . Then \(... | Proof Write \( F \subset G \times X \) for the set of all pairs \( \left( {g, x}\right) \) such that \( {gx} = x \) . The projections \( F \rightarrow X \) and \( F \rightarrow G \) show that\n\n\[ \mathop{\bigsqcup }\limits_{{x \in X}}\operatorname{Stab}\left( x\right) = F = \mathop{\bigsqcup }\limits_{{g \in G}}{X}^{... | Yes |
Given an unrestricted supply of uniform beads of \( n \) distinct colors, how many different necklaces can be made using six beads? | The answer is given by the number of orbits in the natural action of the dihedral group \( {D}_{6} \) on the set of colorings of the dihedral vertices in \( n \) given colors. The group \( {D}_{6} \) consists of 12 transformations: the identity \( e \), two rotations \( {\tau }^{\pm 1} \) by angles \( \pm {60}^{ \circ ... | Yes |
Corollary 12.3 The order of each element of a finite group divides the order of the group. | Proof The order of an element \( g \) is equal to the order of the cyclic subgroup \( \langle g\rangle \) spanned by \( g \) . | No |
In Proposition 12.1 on p.289, we proved that \( g\left( {\ker \varphi }\right) = \left( {\ker \varphi }\right) g \) for every group homomorphism \( \varphi : {G}_{1} \rightarrow {G}_{2} \) and \( g \in {G}_{1} \) . Hence, \( \ker \varphi \vartriangleleft {G}_{1} \) is normal in \( {G}_{1} \) . | This can be seen as well from Exercise 12.25: for every \( h \in \ker \varphi \) and \( g \in G \), we have \[ \varphi \left( {{gh}{g}^{-1}}\right) = \varphi \left( g\right) \varphi \left( h\right) \varphi {\left( g\right) }^{-1} = \varphi \left( g\right) \varphi {\left( g\right) }^{-1} = e. \] Hence \( g\ker \left( \v... | No |
The inner automorphisms of a group \( G \) form a normal subgroup \( \operatorname{Int}\left( G\right) \vartriangleleft \operatorname{Aut}\left( G\right) \) | because for an inner automorphism \( {\operatorname{Ad}}_{g} : h \mapsto {gh}{g}^{-1} \) and an arbitrary automorphism \( \varphi : G \simeq G \), the conjugate automorphism \( \varphi \circ {\operatorname{Ad}}_{g} \circ {\varphi }^{-1} = {\operatorname{Ad}}_{\varphi \left( g\right) } \) is inner. | Yes |
Proposition 12.4 The group structure on \( G/H \) is well defined by (12.17) if and only if \( H \vartriangleleft G \) . | Proof Let formula (12.17) provide \( G/H \) with a well-defined binary operation. Then this operation is associative, because \( \left( {{g}_{1}H \cdot {g}_{2}H}\right) \cdot {g}_{3}H = \left( {{g}_{1}{g}_{2}}\right) H \cdot {g}_{3}H = \) \( \left( {\left( {{g}_{1}{g}_{2}}\right) {g}_{3}}\right) H = \left( {{g}_{1}\lef... | Yes |
Corollary 12.4 (Decomposition of Homomorphisms) A group homomorphism \( \varphi : {G}_{1} \rightarrow {G}_{2} \) can be factored as the composition of a quotient epimorphism\n\n\[ \n{G}_{1} \rightarrow {G}_{1}/\ker \varphi \n\]\n\nfollowed by an injective homomorphism\n\n\[ \n{G}_{1}/\ker \varphi \hookrightarrow {G}_{2... | Proof The corollary states that \( {\varphi }^{-1}\left( {\varphi \left( g\right) }\right) = g\ker \varphi \) for every \( \varphi \left( g\right) \in \operatorname{im}\varphi \) . We have already seen this in Proposition 12.1 on p. 289. | No |
Proposition 12.5 For two subgroups \( N, H \subset G \) such that \( N \vartriangleleft G \), the product \( {HN}\overset{\text{ def }}{ = }\{ {hx} \mid h \in H, x \in N\} \) is a subgroup of \( G \) . Moreover, \( H \cap N \vartriangleleft H, N \vartriangleleft {HN} \) , and \( {HN}/N \simeq H/\left( {H \cap N}\right)... | Proof The product \( {HN} \subset G \) is a subgroup, because for all \( {h}_{1},{h}_{2}, h \in H \) and all \( {x}_{1},{x}_{2}, x \in N \), we have\n\n\[ \n{h}_{1}{x}_{1}{h}_{2}{x}_{2} = \left( {{h}_{1}{h}_{2}}\right) \left( {{h}_{2}^{-1}{x}_{1}{h}_{2} \cdot {x}_{2}}\right) \in {HN}, \n\]\n\n(12.18)\n\n\[ \n{\left( hx... | Yes |
Proposition 13.1 (Universal Property of Free Groups) The map \( i : X \rightarrow {F}_{X} \) that sends \( x \in X \) to the class of the word \( x \) possesses the following universal property: for every group \( G \) and every map of sets \( \Gamma : X \rightarrow G \), there exists a unique group homomorphism \( {\o... | Proof The homomorphism \( {\mathrm{{ev}}}_{\Gamma } \) is unique, because it has to act by the rule described in the proposition. At the same time, this rule certainly produces a well-defined homomorphism \( {\operatorname{ev}}_{\Gamma } : {F}_{X} \rightarrow G \) such that \( \Gamma = {\operatorname{ev}}_{\Gamma } \ci... | Yes |
Proposition 13.2 Let a group \( G \) be generated by elements \( {\left\{ {g}_{x}\right\} }_{x \in X} \) with relating set \( R \subset {F}_{X} \) . Then for every group \( H \) and every family of elements \( {\left\{ {h}_{x}\right\} }_{x \in X} \subset H \) , there exists at most one homomorphism \( \psi : G \rightar... | Proof The family \( {\left\{ {h}_{x}\right\} }_{x \in X} \subset H \) is the same as the map \( \Phi : X \rightarrow H, x \mapsto {h}_{x} \) . By Proposition 13.1, such maps are in bijection with the group homomorphisms \( {\operatorname{ev}}_{\Phi } : {F}_{X} \rightarrow H \) . Such a homomorphism is factorized throug... | Yes |
Theorem 13.2 The groups \( {A}_{n} \) are simple for all \( n \geq 5 \) . | Proof By induction on \( n \) . Let \( N \vartriangleleft {A}_{n} \) . The stabilizer \( {\operatorname{Stab}}_{{A}_{n}}\left( k\right) \) of an element \( k \) is clearly isomorphic to \( {A}_{n - 1} \) . By the induction hypothesis, the subgroup\n\n\[ N \cap {\operatorname{Stab}}_{{A}_{n}}\left( k\right) \vartriangle... | Yes |
Example 13.2 \( \left( {{D}_{n} \simeq \mathbb{Z}/\left( n\right) \rtimes \mathbb{Z}/\left( 2\right) }\right) \) The dihedral group \( {D}_{n} \) contains the normal subgroup of rotations, which is isomorphic to the additive group \( \mathbb{Z}/\left( n\right) \) . The cyclic group of order 2 spanned by a reflection is... | Thus, \( {D}_{n} = \mathbb{Z}/\left( n\right) \rtimes \mathbb{Z}/\left( 2\right) \), and in terms of residue class pairs \( \left( {x, y}\right) \in \mathbb{Z}/\left( n\right) \times \mathbb{Z}/\left( 2\right) \), composition in \( {D}_{n} \) is described by the formula \( \left( {{x}_{1},{y}_{1}}\right) \cdot \left( {... | Yes |
Example 13.3 \( \left( {\operatorname{Aff}\left( V\right) \simeq V \rtimes \operatorname{GL}\left( V\right) }\right) \) We have seen in Example 12.21 on p. 301 that the group \( \operatorname{Aff}\left( V\right) \) of affine automorphisms of the affinization \( \mathbb{A}\left( V\right) \) of a vector space \( V \) con... | By Exercise 12.27 on p. 301, the adjoint action of \( \mathrm{{GL}}\left( V\right) \) on shifts coincides with the tautological action of linear maps on vectors. Therefore, in terms of the pairs \( \left( {v, F}\right) \in V \times \mathrm{{GL}}\left( V\right) \), the composition in \( \operatorname{Aff}\left( V\right)... | No |
Proposition 13.5 Every action of a p-group \( G \) on a finite set \( X \) such that \( p + \left| X\right| \) has a fixed point. | Proof Since the length of every orbit cannot be divisible by \( p \), there is some orbit of length 1 . | No |
Proposition 13.6 Every p-group \( G \) has a nontrivial center | \[ Z\left( G\right) = \{ c \in G \mid \forall g \in {Gcg} = {gc}\} . \] Proof The center \( Z\left( G\right) \) is the fixed-point set of the adjoint action \( {}^{26} \) of \( G \) on itself. Since the lengths of all orbits in \( G \smallsetminus Z\left( G\right) \) are divisible by \( p \), it follows that \( \left| ... | Yes |
Theorem 13.3 (Sylow’s Theorem) For every finite group \( G \) and prime divisor \( p \) of \( \left| G\right| \) there exists a Sylow p-subgroup in \( G \) . Every p-subgroup of \( G \) is contained in some Sylow p-subgroup. All Sylow p-subgroups are conjugate. | Proof Let \( \left| G\right| = {p}^{n}m \), where \( \operatorname{GCD}\left( {m, p}\right) = 1 \) as above. Write \( \mathcal{E} \) for the set of all subsets of cardinality \( {p}^{n} \) in \( G \) . The group \( G \) acts on \( \mathcal{E} \) by left multiplication: an element \( g \in G \) maps \( X \mapsto {gX} = ... | Yes |
Corollary 13.1 (Addendum to Sylow’s Theorem) \( {N}_{p} \mid m \) and \( {N}_{p} \equiv 1\left( {\;\operatorname{mod}\;p}\right) \) for every prime divisor \( p \) of \( \left| G\right| \) . | Proof Write \( \mathcal{S} \) for the set of all Sylow \( p \) -subgroups in \( G \) and consider the adjoint action of \( G \) on \( \mathcal{S} \), in which \( g \in G \) maps \( H \mapsto {gH}{g}^{-1} \) . This action is transitive by Sylow’s theorem. Therefore, \( \left| \mathcal{S}\right| = \left| G\right| /\left|... | Yes |
Example 13.4 (Groups of Order \( {pq} \) for \( \operatorname{GCD}\left( {p - 1, q}\right) = 1 \) ) Let \( \left| G\right| = {pq} \) , where \( p > q \) and both \( p, q \) are prime. Then \( G \) has exactly one Sylow \( p \) -subgroup \( {H}_{p} \simeq \mathbb{Z}/\left( p\right) \), and it is normal. Every Sylow \( q... | \[ {H}_{p} \times {H}_{q} \rightarrow {H}_{p}{H}_{q} \subset G \] is injective, and therefore \( {H}_{p}{H}_{q} = G \) by a cardinality argument. We conclude that \( {H}_{q} \) is complementary to \( {H}_{p} \) . By Sect. 13.4, this forces \[ G = {H}_{p} \rtimes {H}_{q} \simeq \mathbb{Z}/\left( p\right) { \rtimes }_{\p... | Yes |
Example 13.5 (Groups of Order \( {2p} \) ) Let \( \left| G\right| = {2p} \) for prime \( p > 2 \) . The same arguments as in the previous example show that \( G = \mathbb{Z}/\left( p\right) { \rtimes }_{\psi }\mathbb{Z}/\left( 2\right) \) for some action \( \psi : \mathbb{Z}/\left( 2\right) \rightarrow \operatorname{Au... | There are exactly two such elements: \( \psi \left( \left\lbrack 1\right\rbrack \right) = 1 \) and \( \psi \left( \left\lbrack 1\right\rbrack \right) = - 1 \) . For the first choice, the action \( \psi \) is trivial, and \( G \simeq \mathbb{Z}/\left( p\right) \oplus \mathbb{Z}/\left( 2\right) \) . In the second case, t... | Yes |
Lemma 14.1 A subset \( E \subset M \) is a basis of a \( K \) -module \( M \) if and only if for every \( K \) -module \( N \) and every map of sets \( \varphi : E \rightarrow N \), there exists a unique homomorphism of \( K \) -modules \( {f}_{\varphi } : M \rightarrow N \) that extends \( \varphi \) . | Proof If \( E \) is a basis, then every \( K \) -linear map \( {f}_{\varphi } : M \rightarrow N \) that maps \( e \mapsto \varphi \left( e\right) \) should take \( f\left( {\sum e \in E{x}_{e}e}\right) = \sum e \in E{x}_{e}\varphi \left( e\right) \) . On the other hand, the prescription \( {\left( {x}_{e}\right) }_{e \... | Yes |
Every commutative ring \( K \) is clearly a \( K \) -module. A subset \( I \subset K \) is a submodule if and only if \( I \) is an ideal of \( K \) . A principal ideal \( \left( a\right) \subset K \) is a free \( K \) -module if and only if the generator \( a \) is not a zero divisor. | Every nonprincipal ideal is generated by at least two elements, which are linearly related, because any two elements \( a, b \in K \) are linearly related, e.g., as \( a \cdot b - b \cdot a = 0 \) . For example, the ideal \( I = \left( {x, y}\right) \subset \mathbb{Q}\left\lbrack {x, y}\right\rbrack \) considered as a ... | Yes |
Proposition 14.1 If the vectors \( {w}_{1},{w}_{2},\ldots ,{w}_{m} \) span the module \( M \), then for every collection of vectors \( {u}_{1},{u}_{2},\ldots ,{u}_{m} \) in a \( K \) -module \( N \), there exists at most one homomorphism \( F : M \rightarrow N \) such that \( F\left( {w}_{i}\right) = {u}_{i} \) . It ex... | Proof The collection of vectors \( {u}_{1},{u}_{2},\ldots ,{u}_{m} \in N \) is the same as the map\n\n\[ \left\{ {{e}_{1},{e}_{2},\ldots ,{e}_{m}}\right\} \rightarrow N \]\n\nfrom the standard basis of \( {K}^{m} \) to \( N \) . By Lemma 14.1, such maps are in bijection with the \( K \) -linear maps \( {K}^{m} \rightar... | Yes |
Proposition 14.2 Let a \( K \) -module \( M \) be generated by the vectors \( \left( {{w}_{1},{w}_{2},\ldots ,{w}_{m}}\right) \) and suppose the linear endomorphism \( F : V \rightarrow V \) sends them to\n\n\[ \left( {F{w}_{1}, F{w}_{2},\ldots, F{w}_{n}}\right) = \left( {{w}_{1},{w}_{2},\ldots ,{w}_{m}}\right) \cdot {... | Proof Multiplication by \( \det {F}_{w} \) acts on the generating vectors by the rule\n\n\[ \left( {{w}_{1},{w}_{2},\ldots ,{w}_{m}}\right) \mapsto \left( {{w}_{1},{w}_{2},\ldots ,{w}_{m}}\right) \cdot \det {F}_{w} \cdot E = \left( {{w}_{1},{w}_{2},\ldots ,{w}_{m}}\right) \cdot {F}_{w} \cdot {F}_{w}^{ \vee }, \]\n\nwhe... | Yes |
Example 14.3 (Another Proof of the Cayley-Hamilton Identity) Recall that for \( A \in \) \( {\operatorname{Mat}}_{n}\left( K\right) \) and \( f\left( t\right) = {f}_{0} + {f}_{1}t + \cdots + {f}_{m}{t}^{m} \in K\left\lbrack x\right\rbrack \), we write\n\n\[ f\left( A\right) \overset{\text{ def }}{ = }{f}_{0}E + {f}_{1}... | The standard basis vectors \( {e}_{1},{e}_{2},\ldots ,{e}_{n} \) of \( {K}^{n} \) span \( {K}^{n} \) over \( K\left\lbrack t\right\rbrack \) as well. However, over \( K\left\lbrack t\right\rbrack \), they are linearly related. In particular, the homothety with coefficient \( t \) : \( v \mapsto {tv} \) has two differen... | Yes |
Lemma 14.2 Given two submodules \( L, N \subset M \), then \( M = L \oplus N \) if and only if \( L \cup N \) spans \( M \) and \( L \cap N = 0 \) . | Proof \( L \cup N \) spans \( M \) if and only if the addition map\n\n\[ \sigma : L \oplus N \rightarrow M,\left( {a, b}\right) \mapsto a + b, \]\n\nis surjective. The kernel of the addition map is zero if and only if \( L \cap N = 0 \), because \( \left( {a, b}\right) \in \ker \sigma \) if and only if \( a = - b \in L... | Yes |
Theorem 14.1 Let \( M \) be a free module over a commutative ring \( K \) with unit. Then all bases in \( M \) have the same cardinality. | Proof Choose a maximal ideal \( {}^{6}\mathfrak{m} \subset K \) and consider the quotient module \( M/\mathfrak{m}M \) as a vector space over the field \( \mathbb{k} = K/\mathfrak{m} \) . If a set \( E \subset M \) is a basis of \( M \) over \( K \) , then\n\n\[ M = {\bigoplus }_{e \in E}K \cdot e,\;\mathfrak{m}M = {\b... | Yes |
Let us show that \( \mathbb{Z} \) is an indecomposable \( \mathbb{Z} \) -module. | The proper submodules \( L \subset \mathbb{Z} \) are exhausted by the principal ideals \( L = \left( d\right) \) . If there is a submodule \( N \subset \mathbb{Z} \) such that \( \mathbb{Z} = \left( d\right) \oplus N \), then \( N \simeq \mathbb{Z}/\left( d\right) \) by Exercise 14.13. Since \( \mathbb{Z} \) is torsion... | No |
Lemma 14.3 Let \( M \) be a free module of rank \( m < \infty \) over an arbitrary principal ideal domain \( K \) . Then every submodule \( N \subset M \) is free, and \( \operatorname{rk}N \leq \operatorname{rk}M \) . | Proof By induction on \( m = \operatorname{rk}M \) . If \( m = 1 \), then \( M \simeq K \), and the submodule \( N \subset M \) is the principal ideal \( \left( d\right) \subset K \) . If \( d = 0 \), then \( N = 0 \) is free of rank 0 . If \( d \neq 0 \), then \( \left( d\right) = d \cdot K \) is free of rank 1 with b... | Yes |
Lemma 14.4 The numbers \( {\Delta }_{k}\left( C\right) \in K \) (considered up to multiplication by invertible elements of \( K \) ) are not changed under multiplication of \( C \) by an invertible matrix from either side. | Proof Since \( {\Delta }_{k}\left( C\right) = {\Delta }_{k}\left( {C}^{t}\right) \), it is enough to consider left multiplication. Let \( F = \) \( {AC} \), where \( A \) is invertible. Since the rows of \( F \) are \( K \) -linear combinations of the rows of \( C \), every order- \( k \) minor in \( F \) is a \( K \) ... | Yes |
Lemma 14.5 For every pair of elements \( \left( {p, q}\right) \) in the same row (respectively column) of a matrix \( A \) and such that \( p \mid q \) and \( q \mid p \), there exists a generalized Gaussian column operation (respectively row operation) that transforms \( \left( {p, q}\right) \) to \( \left( {d,0}\righ... | Proof Write \( d = \operatorname{GCD}\left( {p, q}\right) \) as \( d = {px} + {qy} \). Then \( p = {ad}, q = {bd} \) for some \( a, b \in K \), and we have the equalities \( {bp} = {aq} \) and \( {ax} + {by} = 1 \), which imply that\n\n\[ \left( {p, q}\right) \cdot \left( \begin{array}{ll} x & - b \\ y & a \end{array}\... | Yes |
Proposition 14.3 Every \( m \times n \) matrix \( C \) over an arbitrary principal ideal domain \( K \) can be transformed to Smith normal form (14.5) by means of generalized Gaussian operations on rows and columns. | Proof After an appropriate permutation of rows and columns, we may assume that \( {c}_{11} \neq 0 \) . If all elements of \( C \) are divisible by \( {c}_{11} \), then the usual elementary row and column operations allow us to eliminate the first column and the first row of \( C \) outside the upper left cell. Inductio... | Yes |
Example 14.5 (Abelian Subgroups in \( {\mathbb{Z}}^{m} \) ) By Lemma 14.3, every abelian subgroup \( L \subset {\mathbb{Z}}^{m} \) is a free \( \mathbb{Z} \) -module. Let \( \operatorname{rk}L = \ell \) . By Theorem 14.2, there exists a basis \( {u}_{1},{u}_{2},\ldots ,{u}_{m} \) in \( {\mathbb{Z}}^{m} \) such that som... | \[ {\mathbb{Z}}^{m}/L \simeq \frac{\mathbb{Z}}{\left( {m}_{1}\right) } \oplus \cdots \oplus \frac{\mathbb{Z}}{\left( {m}_{\ell }\right) } \oplus {\mathbb{Z}}^{m - \ell }. \] | Yes |
Example 14.6 (Commensurable Lattices) An abelian subgroup \( L \subset {\mathbb{Z}}^{m} \) satisfying the equivalent conditions from Exercise 14.19 is called commensurable with \( {\mathbb{Z}}^{m} \) . If \( L \) is given as the \( \mathbb{Z} \) -linear span of columns of some integer rectangular matrix \( C \) , then ... | Note that this can be checked by the standard Gaussian elimination over \( \mathbb{Q} \) as in Sect. 8.4 on p. 182. | No |
Proposition 14.4 Let an abelian subgroup \( L \subset {\mathbb{Z}}^{n} \) be spanned by the columns of a square matrix \( C \in {\operatorname{Mat}}_{n}\left( \mathbb{Z}\right) \) . Then \( L \) is commensurable with \( {\mathbb{Z}}^{n} \) if and only if \( \det C \neq 0 \) . In this case, \( \left| {{\mathbb{Z}}^{n}/L... | Proof Choose a basis \( {u}_{1},{u}_{2},\ldots ,{u}_{n} \) in \( {\mathbb{Z}}^{n} \) such that some multiples\n\n\[ \n{m}_{1}{u}_{1},{m}_{2}{u}_{2},\ldots ,{m}_{\ell }{u}_{\ell }\n\]\n\nform a basis in \( L \) . We have seen in Sect. 14.2.2 that the \( n \times n \) diagonal matrix \( D \) with \( {d}_{ii} = {m}_{i} \)... | Yes |
Lemma 14.6 The correspondence described above establishes a bijection between the finite sequences \( {f}_{1},{f}_{2},\ldots ,{f}_{n} \in K \), where each \( {f}_{i} \) is considered up to multiplication by invertible elements of \( K \) and \( {f}_{i} \mid {f}_{j} \) for all \( i < j \), and the finite unordered colle... | Proof We have to show that every sequence of invariant factors \( {f}_{1},{f}_{2},\ldots ,{f}_{n} \) is uniquely recovered from the collection of its elementary divisors. For every prime \( p \in K \), write \( m\left( p\right) \) and \( n\left( p\right) \), respectively, for the maximal exponent of \( p \) and for the... | Yes |
Example 14.7 The following collection of integer elementary divisors,\n\n\[ \n{3}^{2}{3}^{2}{333} \]\n\n\[ \n{2}^{3}{2}^{3}{2}^{2}2 \]\n\n\[ \n{7}^{2}{77} \]\n\n\[ \n\\text{5 5} \]\n\nappears from the following sequence of invariant factors:\n\n\[ \n{f}_{1} = 3,{f}_{2} = 3 \\cdot 2,{f}_{3} = 3 \\cdot {2}^{2} \\cdot 7,{... | Similarly, looking at the elementary divisors considered before Lemma 14.6,\n\n\[ \n{3}^{3}{333} \]\n\n\[ \n{2}^{2}{2}^{2}{22} \]\n\nwe recover the initial sequence \( 6 = 3 \\cdot 2,6 = 3 \\cdot 2,{12} = 3 \\cdot {2}^{2},{108} = {3}^{3} \\cdot {2}^{2} \) of\n\ninvariant factors. | No |
Theorem 14.4 (Elementary Divisors Theorem) Every finitely generated module M over an arbitrary principal ideal domain \( K \) is isomorphic to\n\n\[ \n{K}^{{n}_{0}} \oplus \frac{K}{\left( {p}_{1}^{{n}_{1}}\right) } \oplus \frac{K}{\left( {p}_{2}^{{n}_{2}}\right) } \oplus \cdots \oplus \frac{K}{\left( {p}_{\alpha }^{{n}... | We split the proof of Theorem 14.4 into several steps presented in Sects. 14.3.2- 14.3.5 below.\n\n## 14.3.2 Existence of the Canonical Decomposition\n\nLet the vectors \( {w}_{1},{w}_{2},\ldots ,{w}_{m} \) span \( M \). Then \( M = {K}^{m}/R \), where \( R \subset {K}^{m} \) is the kernel of the \( K \)-linear surject... | No |
Theorem 14.5 Every finitely generated abelian group is isomorphic to a direct product of additive groups | \[ {\mathbb{Z}}^{r} \oplus \frac{\mathbb{Z}}{\left( {p}_{1}^{{n}_{1}}\right) } \oplus \frac{\mathbb{Z}}{\left( {p}_{2}^{{n}_{2}}\right) } \oplus \cdots \oplus \frac{\mathbb{Z}}{\left( {p}_{\alpha }^{{n}_{\alpha }}\right) }, \] (14.10) where \( {p}_{v},{n}_{i} \in \mathbb{N} \), all \( {p}_{v} \) are prime, and repeated... | Yes |
Theorem 15.1 Each linear operator on a finite-dimensional vector space over an arbitrary field \( \mathbb{k} \) is similar to multiplication by \( t \) in a direct sum of residue modules\n\n\[ \frac{\mathbb{k}\left\lbrack t\right\rbrack }{\left( {p}_{1}^{{m}_{1}}\right) } \oplus \cdots \oplus \frac{\mathbb{k}\left\lbra... | Proof By the elementary divisors theorem, Theorem 14.4 on p. 352, the \( \mathbb{k}\left\lbrack t\right\rbrack \) -module \( {V}_{F} \) associated with a arbitrary operator \( F : V \rightarrow V \) is a direct sum of a free module \( \mathbb{k}{\left\lbrack t\right\rbrack }^{m} \) and a torsion module of the form (15.... | Yes |
Each indecomposable operator is similar to multiplication by \( t \) in the residue module \( \mathbb{k}\left\lbrack t\right\rbrack /\left( {p}^{m}\right) \), where \( p \in \mathbb{k}\left\lbrack t\right\rbrack \) is monic irreducible. Such an operator is irreducible if and only if \( m = 1 \) . Two indecomposable ope... | The first and third statements are contained in Theorem 15.1. Let us prove the second. Every \( \mathbb{k}\left\lbrack t\right\rbrack \) -submodule in \( \mathbb{k}\left\lbrack t\right\rbrack /\left( {p}^{m}\right) \) is an ideal of this quotient ring. For \( m = 1 \), the residue ring \( \mathbb{k}\left\lbrack t\right... | Yes |
Lemma 15.1 Write the elements of \( \mathbb{k}{\left\lbrack t\right\rbrack }^{n} \) as coordinate columns with elements in \( \mathbb{k}\left\lbrack t\right\rbrack \) . Then the relation submodule \( \ker {\pi }_{v} \subset \mathbb{k}{\left\lbrack t\right\rbrack }^{n} \) is spanned over \( \mathbb{k}\left\lbrack t\righ... | Proof Let \( {F}_{v} = \left( {f}_{ij}\right) \) . Then the \( j \) th column of \( {tE} - {F}_{v} \) is expressed through the standard basis \( e \) as \( t{e}_{j} - {f}_{1j}{e}_{1} - {f}_{2j}{e}_{2} - \cdots - {f}_{nj}{e}_{n} \) . When we apply \( {\pi }_{v} \) to this vector, we get\n\n\[ \n{\pi }_{v}\left( {t{e}_{j... | No |
Proposition 15.1 Every operator \( F \) over a field \( \mathbb{R} \) has an invariant subspace of dimension \( \leq 2 \) . | Proof Let the characteristic polynomial of \( F \) be factored as \( {\chi }_{F} = {q}_{1} \cdot {q}_{2}\cdots {q}_{m} \) , where \( {q}_{i} \in \mathbb{R}\left\lbrack t\right\rbrack \) are monic irreducible polynomials, not necessarily distinct. Note that \( \deg {q}_{i} \leq 2 \) for each \( i \) . If we apply the ze... | Yes |
Proposition 15.2 The following properties of an operator \( F : V \rightarrow V \) are equivalent:\n\n(1) \( V \) is a direct sum of irreducible \( F \) -invariant subspaces.\n\n(2) \( V \) is linearly generated by irreducible \( F \) -invariant subspaces.\n\n(3) For every proper \( F \) -invariant subspace \( U \subse... | Proof The implication \( \left( 1\right) \Rightarrow \left( 2\right) \) is obvious. Let us prove the implication \( \left( 2\right) \Rightarrow \) (3). For every irreducible \( F \) -invariant subspace \( L \subset V \), the intersection \( L \cap U \) is either zero or \( L \), because \( L \cap U \subset L \) is \( F... | No |
Let us show that every orthogonal operator \( {}^{15} \) \( F : V \rightarrow V \) on a Euclidean vector space \( V \) is semisimple. | For every proper \( F \) - invariant subspace \( U \subset V \), there is an orthogonal decomposition \( {}^{16}V = U \oplus {U}^{ \bot } \) . Let us verify that the orthogonal complement \( {U}^{ \bot } \) is \( F \) -invariant. Since the restriction \( {\left. F\right| }_{U} \) is a Euclidean isometry of \( U \), it ... | Yes |
Theorem 15.2 (Normal Form of a Euclidean Isometry) For every Euclidean isometry \( F : V \rightarrow V \), there exists an orthonormal basis of \( V \) in which \( F \) has a block-diagonal matrix formed by one-cell blocks \( \pm 1 \) and \( 2 \times 2 \) blocks\n\n\[ \left( \begin{matrix} \cos \varphi & - \sin \varphi... | Proof The existence was explained in Example 15.1. To see uniqueness, note that each 1-dimensional isometry \( \pm \mathrm{{Id}} \) contributes an elementary divisor \( t \mp 1 \) to \( \mathcal{E}\left( F\right) \) , and each rotation of the plane by the angle \( \varphi \) contributes an elementary divisor \( {t}^{2}... | Yes |
Proposition 15.3 The following properties of an operator \( F : V \rightarrow V \) are equivalent:\n\n(1) \( F \) is cyclic.\n\n(2) \( F \) is similar to multiplication by \( t \) in the residue class module \( \mathbb{k}\left\lbrack t\right\rbrack /\left( f\right) \), where \( f \in \mathbb{k}\left\lbrack t\right\rbra... | Proof The equivalence (3) \( \Leftrightarrow \) (4) follows at once from Corollary 15.4. Both conditions mean that \( F \) is similar to multiplication by \( t \) in the direct sum of residue class modules\n\n\[ \mathbb{k}\left\lbrack t\right\rbrack /\left( {p}_{1}^{{m}_{1}}\right) \oplus \mathbb{k}\left\lbrack t\right... | Yes |
Proposition 15.4 \( {\operatorname{Spec}}_{\mathbb{k}}\left( F\right) \) coincides with the set of roots \( {}^{17} \) of the characteristic polynomial \( {\chi }_{F}\left( t\right) = \det \left( {t\operatorname{Id} - F}\right) \) in \( \mathbb{k} \) . In particular, \( \left| {\operatorname{Spec}\left( F\right) }\righ... | Proof An eigenvector \( v \) with eigenvalue \( \lambda \) lies in the kernel of the linear operator \( \lambda \mathrm{{Id}} - F \) . By Corollary Exercise 9.12, \( \ker \left( {{\lambda E} - F}\right) \neq 0 \) if and only if \( \det \left( {\lambda \operatorname{Id} - F}\right) = 0 \) . | No |
Proposition 15.5 Every set of eigenvectors with distinct eigenvalues is linearly independent. | Proof Assume the contrary and let \( {x}_{1}{v}_{1} + {x}_{2}{v}_{2} + \cdots + {x}_{k}{v}_{k} = 0 \) be a linear relation of minimal length \( k \) . Write \( {\lambda }_{1},{\lambda }_{2},\ldots ,{\lambda }_{k} \) for the eigenvalues of \( {v}_{1},{v}_{2},\ldots ,{v}_{m} \) respectively. If we apply the operator to t... | Yes |
Proposition 15.6 The following properties of a linear operator \( F : V \rightarrow V \) are equivalent:\n\n(1) \( F \) is diagonalizable.\n\n(2) \( V \) is spanned by the eigenvectors of \( F \) .\n\n(3) The characteristic polynomial \( {\chi }_{F}\left( t\right) = \det \left( {{tE} - F}\right) \) is completely factor... | Proof The equivalences \( \left( 2\right) \Leftrightarrow \left( 1\right) \Leftrightarrow \left( 4\right) \) and the implication \( \left( 1\right) \Rightarrow \left( 3\right) \) are evident from the discussion preceding the proposition. The equivalence (4) \( \Leftrightarrow \) (5) follows at once from Corollary 15.3.... | Yes |
Corollary 15.9 The restriction of a diagonalizable operator \( F \) to an \( F \) -invariant subspace is diagonalizable as well. | Proof This is provided by (5). | No |
Corollary 15.10 Over an algebraically closed field, diagonalizability is equivalent to semisimplicity. | Proof The monic irreducible polynomials over an algebraically closed field are exhausted by the linear binomials \( t - \lambda \) . | No |
Assume that char \( \mathbb{k} \neq 2 \) . A linear operator \( \sigma : V \rightarrow V \) is called an involution if \( {\sigma }^{2} = {\operatorname{Id}}_{V} \) . Thus, \( \sigma \) is annihilated by the polynomial \( {t}^{2} - 1 = \) \( \left( {t + 1}\right) \left( {t - 1}\right) \), which satisfies condition (5) ... | either \( \sigma = \pm {\operatorname{Id}}_{V} \) or \( V = {V}_{ + } \oplus {V}_{ - } \), where\n\n\[ \n{V}_{ + } = \ker \left( {\sigma - {\mathrm{{Id}}}_{V}}\right) = \mathrm{{im}}\left( {\sigma + {\mathrm{{Id}}}_{V}}\right) \;\mathrm{{and}}\;{V}_{ - } = \ker \left( {\sigma + {\mathrm{{Id}}}_{V}}\right) = \mathrm{{im... | Yes |
A linear operator \( \pi : V \rightarrow V \) is called a projector if \( {\pi }^{2} = \pi \). | Thus, \( \pi \) is annihilated by the polynomial \( {t}^{2} - t = t\left( {t - 1}\right) \), which also satisfies condition (5) of Proposition 15.6. We conclude that either \( \pi = 0 \), or \( \pi = \operatorname{Id} \), or \( V = {V}_{0} \oplus {V}_{1} \), where \( {V}_{0} = \ker \pi \) and \( {V}_{1} = \{ v \in V \m... | Yes |
Proposition 15.7 Assume that the operator \( F : V \rightarrow V \) is annihilated by the polynomial \( q \in \mathbb{k}\left\lbrack t\right\rbrack \) factored in \( \mathbb{k}\left\lbrack t\right\rbrack \) as\n\n\[ q = {q}_{1} \cdot {q}_{2}\cdots {q}_{r}\text{, where}\forall i, j\operatorname{GCD}\left( {{q}_{i},{q}_{... | Proof The invariance of \( \ker {q}_{i}\left( F\right) \) is obvious: \( {q}_{i}\left( F\right) v = 0 \Rightarrow {q}_{i}\left( F\right) {Fv} = \) \( F{q}_{i}\left( F\right) v = 0 \) . The inclusion \( \operatorname{im}{Q}_{i}\left( F\right) \subset \ker {q}_{i}\left( F\right) \) follows from the vanishing of \( {q}_{i... | Yes |
Proposition 15.8 Every set of commuting linear operators over an algebraically closed field has a common eigenvector. Over an arbitrary field, every set of diagonalizable commuting operators can be simultaneously diagonalized in a common basis. | Proof By induction on \( \dim V \), where \( V \) is the space on which the operators act. Both statements are obvious if all the operators are scalar dilations, which holds, in particular, for \( \dim V = 1 \) . Assume that one of the operators, call it \( F \), is nonscalar. To prove the first statement, note that \(... | Yes |
Theorem 15.4 Let a subalgebra \( \mathcal{C} \subset {\mathbb{C}}^{\mathbb{C}} \) be suitable for evaluation at an operator \( F : V \rightarrow V \) . Then there exists a unique homomorphism of \( \mathbb{C} \) -algebras \( {\operatorname{ev}}_{F} : \mathcal{C} \rightarrow \operatorname{End}V \) whose restriction to t... | Proof Let \( {\mathrm{{ev}}}_{F} : \mathcal{C} \rightarrow \mathbb{C} \) be a required homomorphism of \( \mathbb{C} \) -algebras. Fix an arbitrary function \( f \in \mathcal{C} \) and evaluate both sides of (15.14) at \( F \) . This leads to the equality\n\n\[ f\left( F\right) = f\left( \lambda \right) \cdot E + {f}^{... | Yes |
The solution of the linear recurrence equation\n\n\[ \n{z}_{k} + {a}_{1}{z}_{k - 1} + {a}_{2}{z}_{k - 2} + \cdots + {a}_{m}{z}_{m - n} = 0 \n\]\n\n(15.18)\n\nof order \( m \) in the unknown sequence \( \left( {z}_{n}\right) \) is equivalent to evaluation of the power function \( t \mapsto {t}^{n} \) on the shifting mat... | \[ \nS = \left( \begin{matrix} 0 & 0 & \cdots & 0 & {\alpha }_{m} \\ 1 & 0 & \ddots & 0 & {\alpha }_{m - 1} \\ 0 & 1 & \ddots & \vdots & \vdots \\ \vdots & \ddots & \ddots & 0 & {\alpha }_{2} \\ 0 & \cdots & 0 & 1 & {\alpha }_{1} \end{matrix}\right) \n\]\n\nwhich has size \( m \times m \) and is uniquely determined by\... | Yes |
The shifting matrix for the Fibonacci \( {\operatorname{numbers}}^{24}{z}_{n} = {z}_{n - 1} + {z}_{n - 2} \) is \( S = \left( \begin{array}{ll} 0 & 1 \\ 1 & 1 \end{array}\right) \). Its characteristic polynomial \( {\chi }_{S}\left( t\right) = \) \( {t}^{2} - t\operatorname{tr}S + \det S = {t}^{2} - t - 1 = \left( {t -... | \[ \left\{ \begin{array}{l} a{\lambda }_{ + } + b = {\lambda }_{ + }^{n} \\ a{\lambda }_{ - } + b = {\lambda }_{ - }^{n} \end{array}\right. \] and they are equal to \( a = \left( {{\lambda }_{ + }^{n} - {\lambda }_{ - }^{n}}\right) /\left( {{\lambda }_{ + } - {\lambda }_{ - }}\right), b = \left( {{\lambda }_{ + }^{n + ... | Yes |
Proposition 15.9 Under the conditions of Theorem 15.4, the spectrum \( \operatorname{Spec}f\left( F\right) \) consists of the numbers \( f\left( \lambda \right) ,\lambda \in \operatorname{Spec}F \) . If \( {f}^{\prime }\left( \lambda \right) \neq 0 \), then the elementary divisors \( {\left( t - \lambda \right) }^{m} \... | Proof Realize \( F \) as multiplication by \( t \) in\n\n\[ V = \frac{\mathbb{C}\left\lbrack t\right\rbrack }{\left( {\left( t - {\lambda }_{1}\right) }^{{s}_{1}}\right) } \oplus \cdots \oplus \frac{\mathbb{C}\left\lbrack t\right\rbrack }{\left( {\left( t - {\lambda }_{r}\right) }^{{s}_{r}}\right) }.\]\n\nThe proof of ... | Yes |
Proposition 16.1 (Nondegeneracy Criteria) Let a bilinear form \( \beta : V \times V \rightarrow \mathbb{k} \) have Gramian \( {B}_{e} \) in some basis \( e = \left( {{e}_{1},{e}_{2},\ldots ,{e}_{n}}\right) \) of \( V \) . The following conditions on \( \beta \) are equivalent:\n\n(1) \( \det {B}_{e} \neq 0 \) .\n\n(2) ... | Proof Conditions (2) and (4) mean that \( \ker {\beta }^{ * } = 0 \) and \( \operatorname{im}{\beta }^{ * } = {V}^{ * } \) respectively. Each of them is equivalent to (3), because \( \dim V = \dim {V}^{ * } \) . For the same reason, conditions (5),(6),(7) are mutually equivalent as well. Since the operators \( \beta ,{... | Yes |
Example 16.2 (hyperbolic Form) A symmetric bilinear form on an even-dimensional coordinate space \( {\mathbb{k}}^{2n} \) with Gramian\n\n\[ H = \left( \begin{array}{ll} 0 & E \\ E & 0 \end{array}\right) \]\n\n(16.13)\n\nin the standard basis is called a hyperbolic form. Here \( E \) and 0 mean the identity and the zero... | Over an algebraically closed field, the hyperbolic form is equivalent to the Euclidean form and admits an orthonormal basis formed by the vectors\n\n\[ {\varepsilon }_{{2v} - 1} = \left( {{e}_{v} - {e}_{n + v}}\right) /\sqrt{-2}\;\text{ and }\;{\varepsilon }_{2v} = \left( {{e}_{v} + {e}_{n + v}}\right) /\sqrt{2},\;1 \l... | Yes |
Proposition 16.2 The dimension of an isotropic subspace \( U \) of an arbitrary nondegenerate bilinear form on \( V \) is bounded above by the inequality. | Proof A subspace \( U \subset V \) is isotropic for \( \beta \) if and only if \( \beta : V \simeq {V}^{ * } \) sends \( U \) to Ann \( U \subset {V}^{ * } \) . Since \( \beta \) is injective by the nondegeneracy assumption, \( \dim U \leq \) \( \dim \operatorname{Ann}U = \dim V - \dim U. | Yes |
Example 16.6 (Nondegenerate Form of Type \( {U}_{n} \) ) For \( n \in \mathbb{N} \), write \( {U}_{n} \) for the coordinate space \( {\mathbb{k}}^{n} \) equipped with a bilinear form \( \beta \) whose Gramian in the standard basis is\n\n\[ B = \\left( \\begin{matrix} & & & & 1 \\ & & & - 1 & 1 \\ & & & 1 & - 1 \\ & & \... | The canonical operator \( K = {B}^{-1}{B}^{t} \) of this form is equal to\n\n\[ \\left( {\\underset{\\begin{matrix} . \\\\ - 1 \\\\ 1 \\\\ 1 \\\\ 1 \\end{matrix}}{.}{1}^{n}{1}^{n}{2}^{n}{2}^{n}{2}^{n}{2}^{n}{2}^{n}{2}^{n}{2}^{n}{2}^{n}{2}^{n}{2}^{n}{2}^{n}{2}^{n}{2}^{n}{2}^{n}{2}^{n}{2}^{n}{2}^{n}{2}^{n}{2}^{n}{2}^{n}{... | Yes |
Proposition 16.3 (Reflexivity) For every vector space \( V \) with nondegenerate bilinear form \( \beta \) and linear operator \( f : V \rightarrow V \), the following properties are equivalent:\n\n(1) \( {f}^{\vee \vee } = f \) ,\n\n(2) \( {}^{\vee \vee }f = f \) ,\n\n(3) \( {}^{ \vee }f = {f}^{ \vee } \) ,\n\n(4) \( ... | Proof By Exercise 16.12, the right adjunction of both sides in (3) leads to (1), from which (3) can be reobtained by left adjunction of both sides. For the same reason, (3) and (2) are equivalent as well. Property (3) can be written as \( {\left( {\beta }^{ * }\right) }^{-1}{f}^{ * }{\beta }^{ * } = \) \( {\beta }^{-1}... | No |
Lemma 16.1 Nondegenerate bilinear forms \( \alpha ,\beta \) on \( V \) have equal canonical operators \( {\varkappa }_{\alpha } = {\varkappa }_{\beta } \) if and only if \( \alpha = {\beta f} \) for some linear operator \( f : V \rightarrow V \) that is self-adjoint with respect to both forms. | Proof Let the canonical operators be equal, i.e., \( {\beta }^{-1}{\beta }^{ * } = {\alpha }^{-1}{\alpha }^{ * } \) . Dual to this equality is \( \beta {\left( {\beta }^{ * }\right) }^{-1} = \alpha {\left( {\alpha }^{ * }\right) }^{-1} \) . The operator \( f = {\beta }^{-1}\alpha = {\left( {\beta }^{ * }\right) }^{-1}{... | Yes |
Theorem 16.1 Let the ground field \( \mathbb{k} \) be algebraically closed of zero characteristic. Then two nondegenerate bilinear forms are equivalent if and only if their canonical operators are similar. | Proof Given a linear automorphism \( g : V \rightarrow V \) such that \( \alpha = {g}^{ * }{\beta g} \), then\n\n\[{\varkappa }_{\alpha } = {\alpha }^{-1}{\alpha }^{ * } = {g}^{-1}{\beta }^{-1}{\beta }^{ * }g = {g}^{-1}{\varkappa }_{\beta }g.\]\n\nConversely, let \( \alpha \) and \( \beta \) have similar canonical oper... | Yes |
Lemma 16.2 Over an algebraically closed field \( \mathbb{k} \) of characteristic zero, for every finite-dimensional nondegenerate operator \( f \) there exists a polynomial \( p\left( t\right) \in \mathbb{k}\left\lbrack t\right\rbrack \) such that \( p{\left( f\right) }^{2} = f \) . | Proof Realize \( f \) as multiplication by \( t \) in the direct sum of residue modules of type \( \mathbb{k}\left\lbrack t\right\rbrack /{\left( t - \lambda \right) }^{m} \), where \( \lambda \neq 0 \) by the assumption of the lemma. For each \( \lambda \), write \( {m}_{\lambda } \) for the maximal exponent of binomi... | Yes |
Proposition 16.4 If \( \dim V < \infty \) and \( \beta \) is nondegenerate, then for every subspace \( U \subset V \) , we have\n\n\[ \dim {}^{ \bot }U = \dim V - \dim U = \dim {U}^{ \bot }\;\text{ and }\;{\left( {}^{ \bot }U\right) }^{ \bot } = U = {}^{ \bot }\left( {U}^{ \bot }\right) . | Proof The first pair of equalities hold because the left and right orthogonals are the preimages of the same space Ann \( U \subset {V}^{ * } \) of dimension \( {}^{24}\dim \operatorname{Ann}U = \dim V - \) \( \dim U \) under the linear isomorphisms \( {\beta }^{ * },\beta : V \simeq {V}^{ * } \) . Since \( U \) is a s... | Yes |
Proposition 16.5 Let \( V \) be an arbitrary \( {}^{25} \) vector space with bilinear form \( \beta \), and let \( U \subset V \) be a finite-dimensional subspace such that the restriction of \( \beta \) to \( U \) is nondegenerate. Then \( V = {U}^{ \bot } \oplus U = U \oplus {}^{ \bot }U \), and for every vector \( v... | Proof For every \( v \in V \), the existence of the decomposition \( v = w + {v}_{U} \) with \( w \in \) \( {U}^{ \bot },{v}_{U} \in U \) means the existence of a vector \( {v}_{U} \in U \) such that \( \beta \left( {u, v}\right) = \beta \left( {u,{v}_{U}}\right) \) for all \( u \in U \), because the latter equality sa... | Yes |
Corollary 16.1 Let \( V \) be a finite-dimensional vector space with nondegenerate bilinear form \( \beta \) and let \( U \subset V \) be a subspace such that the restriction of \( \beta \) to \( U \) is nondegenerate as well. Then the restrictions of \( \beta \) to both orthogonals \( {U}^{ \bot },{}^{ \bot }U \) are ... | Proof For every \( {w}^{\prime \prime } \in {U}^{ \bot } \) there exists some \( v \in V \) such that \( \beta \left( {v,{w}^{\prime \prime }}\right) \neq 0 \) . We use the decomposition \( V = {U}^{ \bot } \oplus U \) from Proposition 16.5 to write it as \( v = \) \( {}_{{U}^{ \bot }}v + {v}_{U} \), where \( {}_{{U}^{... | Yes |
Corollary 16.2 Under the assumptions of Corollary 16.1, the restrictions on \( {U}^{ \bot } \) and \( {}^{ \bot }U \) of the two projections along \( U \) in the decompositions\n\n\[ U \oplus {}^{ \bot }U = V = {U}^{ \bot } \oplus U \]\n\nestablish the inverse isometric isomorphisms\n\n\[ \lambda : {U}^{ \bot } \simeq ... | Proof The projection (16.30) is isometric, because for all \( {w}^{\prime },{w}^{\prime \prime } \in {U}^{ \bot } \), \n\n\[ \beta \left( {{w}_{{ \bot }_{U}}^{\prime },{w}_{{ \bot }_{U}}^{\prime \prime }}\right) = \beta \left( {{w}^{\prime } - {}_{U}{w}^{\prime },{w}^{\prime \prime } - {}_{U}{w}^{\prime \prime }}\right... | Yes |
Lemma 16.3 Let \( f : V \simeq V \) be an isometry of an arbitrary nondegenerate bilinear form \( \beta \) on \( V \) and let\n\n\[ 0 = {u}_{0},{u}_{1},{u}_{2},\ldots ,{u}_{\ell },\;0 = {w}_{0},{w}_{1},{w}_{2},\ldots ,{w}_{m}, \]\n\nbe two Jordan chains for \( f \) with eigenvalues \( \lambda \) and \( \mu \) respectiv... | Proof Since \( \beta \left( {{u}_{i},{w}_{j}}\right) = \beta \left( {\varkappa {u}_{i},\varkappa {w}_{j}}\right) = \beta \left( {\lambda {u}_{i} + {u}_{i - 1},\mu {w}_{j} + {w}_{j - 1}}\right) \), for all \( 1 \leq i \leq \ell \) and \( 1 \leq j \leq m \) we have the following recurrence relation:\n\n\[ \left( {1 - {\l... | Yes |
Theorem 16.2 Let \( f : V \simeq V \) be an isometry of a nondegenerate indecomposable bilinear form \( \beta \) on \( V \) . Then \( \mathcal{E}\ell \left( f\right) \) consists either of \( k \) binomials \( {\left( t - 1\right) }^{m} \) with common \( \begin{matrix} & m = \dim V/k\;\text{or}\;\text{of}k\;\text{binomi... | Proof Fix some Jordan basis \( \mathbf{e} \) for \( f \), and for each \( \lambda \in \operatorname{Spec}f \), write \( {U}_{\lambda } \) for the linear span of all Jordan chains of maximal length with eigenvalue \( \lambda \) in \( \mathbf{e} \) . By Lemma 16.3, \( V \) splits into a direct biorthogonal sum \( v = W \... | Yes |
Theorem 16.3 Over an algebraically closed field of zero characteristic, a finite collection of possibly repeated binomials \( {\left( t - \lambda \right) }^{m} \) with some \( \lambda \in \mathbb{k}, m \in \mathbb{N} \) is realized as a collection of elementary divisors of the canonical operator of a nondegenerate bili... | Proof Let a collection of elementary divisors \( {\left( t - \lambda \right) }^{m} \) satisfy conditions (1)-(4). Then it consists of some disjoint pairs of the form \( {\left( t - \lambda \right) }^{m},{\left( t - {\lambda }^{-1}\right) }^{m} \), where repeated pairs are allowed, and some unpaired divisors \( {\left( ... | Yes |
Proposition 16.6 For every subspace \( U \subset V \), complementary \( {}^{33} \) to \( \ker \beta \) the restriction to \( U \) of the (skew) symmetric form \( \beta \) is nondegenerate. | Proof Let \( u \in U \) satisfy \( \beta \left( {u, w}\right) = 0 \) for all \( w \in U \) . Since \( V = U \oplus \ker \beta \) , we can write every \( v \in V \) as \( v = w + e \), where \( w \in U \) and \( e \in \ker \beta \) . Then \( \beta \left( {u, v}\right) = \beta \left( {u, w}\right) + \beta \left( {u, e}\r... | Yes |
Theorem 16.4 (Lagrange’s Theorem) Let \( \beta \) be a symmetric bilinear form on a finite-dimensional vector space over an arbitrary field \( \mathbb{k} \) with char \( \mathbb{k} \neq 2 \). Then \( \beta \) has a diagonal Gramian in some basis. | Proof Induction on \( \dim V \). If \( \dim V = 1 \) or if \( \beta \) is zero, then the Gramian of \( \beta \) is diagonal in every basis. If \( \beta \neq 0 \), then there exists some vector \( e \in V \) such that \( \beta \left( {e, e}\right) \neq 0 \), because otherwise, \[ \beta \left( {u, w}\right) = \left( {\be... | Yes |
Corollary 16.4 Two symmetric bilinear forms over an algebraically closed field \( \mathbb{k} \) with \( \operatorname{char}\left( \mathbb{k}\right) \neq 2 \) are equivalent if and only if they have equal ranks. | Proof Over an algebraically closed field, every nonzero diagonal element of the Gramian is equal to 1 after dividing the corresponding basis vector \( {e}_{i} \) by \( \sqrt{\left( {e}_{i},{e}_{i}\right) } \). | No |
Theorem 16.5 (Darboux’s Theorem) Over an arbitrary field, \( {}^{35} \) every finite-dimensional vector space \( V \) with nondegenerate skew-symmetric form \( \omega \) is equivalent to a symplectic space. \( {}^{36} \) In particular, \( \dim V \) is necessarily even. | Proof We will construct a basis \( {e}_{1},{e}_{2},\ldots ,{e}_{2n} \) in \( V \) with block diagonal Gramian formed by \( 2 \times 2 \) blocks\n\n\[ \left( \begin{matrix} 0 & 1 \\ - 1 & 0 \end{matrix}\right) \]\n\n(16.33)\n\nAfter that, we reorder the basis vectors by writing first all vectors \( {e}_{i} \) with odd \... | Yes |
Proposition 16.7 Every isotropic subspace \( U \subset V \) is contained in some symplectic subspace \( W \subset V \) of dimension \( \dim W = 2\dim U \) . Every basis of \( U \) can be extended to some symplectic basis of \( W \) . | Proof Chose a basis \( {u}_{1},{u}_{2},\ldots ,{u}_{m} \) in \( U \), extend it to some basis in \( V \), and write \( {u}_{1}^{ \vee },{u}_{2}^{ \vee },\ldots ,{u}_{m}^{ \vee } \) for the first \( m \) vectors of the dual basis with respect to \( \omega \) . Therefore,\n\n\[ \n\omega \left( {{u}_{i},{u}_{j}^{ \vee }}\... | Yes |
Theorem 16.6 For every Lagrangian subspace \( L \subset V \), there exists a Lagrangian subspace \( {L}^{\prime } \subset V \) such that \( V = L \oplus {L}^{\prime } \) . For every basis \( e \) in \( L \), there exists a unique basis \( {\mathbf{e}}^{\prime } \) in \( {L}^{\prime } \) such that the 2n vectors \( \mat... | Proof If we repeat the proof of Proposition 16.7 for \( U = L \) and \( \left( {{u}_{1},{u}_{2},\ldots ,{u}_{m}}\right) = \) \( \mathbf{e} \), then we get \( W = V = L \oplus {L}^{\prime } \), where \( {L}^{\prime } \) is spanned by the vectors \( {w}_{j} \) from (16.39).\n\nThis proves the first statement. To prove th... | Yes |
Problem 16.17 (Markov’s Equation) Let \( {\beta }_{\mathbb{C}} : {\mathbb{C}}^{3} \times {\mathbb{C}}^{3} \rightarrow \mathbb{C} \) be a nonsymmetric indecomposable bilinear form such that its restriction to \( {\mathbb{Z}}^{3} \subset {\mathbb{C}}^{3} \) takes integer values and has Gram determinant 1 in the standard ... | Hint: since \( \left( {{\mathbb{C}}^{3},{\beta }^{\mathbb{C}}}\right) \) is isomorphic to \( {U}_{3} \), the canonical operator of \( \beta \) has \( \operatorname{tr}{B}_{u}^{-1}{B}_{u}^{t} = 3 \) . | No |
Example 17.1 (Binary Quadratic Forms) Quadratic forms in two variables are called binary. Every nonzero binary quadratic form | \[ q\left( x\right) = a{x}_{1}^{2} + {2b}{x}_{1}{x}_{2} + c{x}_{2}^{2} = \left( {{x}_{1},{x}_{2}}\right) \left( \begin{array}{ll} a & b \\ b & c \end{array}\right) \left( \begin{array}{l} {x}_{1} \\ {x}_{2} \end{array}\right) \] in appropriate coordinates \( {t}_{1},{t}_{2} \) becomes either \( \alpha {t}_{1}^{2} \), w... | Yes |
Proposition 17.1 Let \( V \) be a vector space with nondegenerate symmetric bilinear form \( \beta \) . Then every isotropic subspace \( U \subset V \) is contained in some hyperbolic subspace \( W \subset V \) of dimension \( \dim W = 2\dim U \) . Every basis of \( U \) can be extended to a hyperbolic basis of \( W \)... | Proof Choose a basis \( {u}_{1},{u}_{2},\ldots ,{u}_{m} \) in \( U \), extend it to a basis in \( V \), and write \( {u}_{1}^{ \vee },{u}_{2}^{ \vee },\ldots ,{u}_{m}^{ \vee } \) for the first \( m \) vectors of the dual basis with respect to \( \beta \) . Therefore,\n\n\[ \beta \left( {{u}_{i},{u}_{j}^{ \vee }}\right)... | Yes |
Theorem 17.1 Every vector space \( V \) with nondegenerate symmetric bilinear form is an orthogonal direct sum \( V = {H}_{2k} \oplus U \), where \( {H}_{2k} \subset V \) is some hyperbolic subspace, \( U = {H}_{2k}^{ \bot } \) is anisotropic, and either of the two summands may vanish. | Proof Induction on \( \dim V \) . If \( V \) is anisotropic (in particular, for \( \dim V = 1 \) ), there is nothing to prove. If there exists some isotropic nonzero vector \( e \in V \), then by Proposition 17.1, \( e \) is contained in some hyperbolic plane \( {H}_{2} \) . Since the form is nondegenerate in this plan... | Yes |
Example 17.2 (Isometries of the Hyperbolic Plane) Let the linear operator \( f \) : \( {H}_{2} \rightarrow {H}_{2} \) have matrix \( F = \left( \begin{array}{ll} a & b \\ c & d \end{array}\right) \) in a hyperbolic basis \( e,{e}^{ * } \) of \( {H}_{2} \) . Then \( F \) is orthogonal if and only if | \n\[
\left( \begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right) = \left( \begin{array}{ll} a & c \\ b & d \end{array}\right) \cdot \left( \begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right) \cdot \left( \begin{array}{ll} a & b \\ c & d \end{array}\right) = \left( \begin{matrix} {2ac} & {ad} + {bc} \\ {ad} + {bc} & {2bd... | Yes |
Lemma 17.1 In a space with a nonsingular quadratic form \( q \), for every pair of anisotropic vectors \( u, w \) such that \( q\left( u\right) = q\left( w\right) \), there exists a reflection that sends \( u \) either to w or to \( - w \) . | Proof If \( u, w \) are collinear, then the reflection \( {\sigma }_{w} = {\sigma }_{u} \) sends them to each other. Let \( u, w \) span the 2-dimensional plane \( U \) . The two diagonals of the rhombus spanned by \( u, v \) are \( u + w \) and \( u - v \) . They are perpendicular: \( \widetilde{q}\left( {u + v, u - v... | Yes |
Theorem 17.2 Every isometry of an n-dimensional space with a nonsingular symmetric form is a composition of at most \( {2n} \) reflections in hyperplanes. | Proof Induction on \( n \) . For \( n = 1 \), the orthogonal group of the anisotropic line consists of the identity \( E \) and reflection \( - E \), because the linear map \( v \mapsto {\lambda v} \) satisfying \( q\left( v\right) = q\left( {\lambda v}\right) = {\lambda }^{2}q\left( v\right) \) has \( \lambda = \pm 1 ... | Yes |
Lemma 17.2 (Witt’s Lemma) Let \( U, V, W \) be spaces with nonsingular symmetric bilinear forms. If the orthogonal direct sum \( U \oplus V \) is isometrically isomorphic to the orthogonal direct sum \( U \oplus W \), then \( V \) and \( W \) are isometrically isomorphic. | Proof Induction on \( \dim U \) . If \( \dim U = 1 \), then \( U = \mathbb{k} \cdot u \), where \( u \) is anisotropic. Write\n\n\[ f : \mathbb{k} \cdot u \oplus V \simeq \mathbb{k} \cdot u \oplus W \]\n\nfor the isometric isomorphism in question. Let \( \sigma \) be the reflection of the second space such that \( \sig... | Yes |
Corollary 17.2 Let \( V \) be a space with a nonsingular symmetric bilinear form. Then in the orthogonal decomposition \( V = {H}_{2k} \oplus U \) from Theorem 17.1 on p. 426, the hyperbolic space \( {H}_{2k} \) and anisotropic space \( U \) are determined by \( V \) uniquely up to isometry. In other words, given two s... | Proof Let \( m \geq k \), that is, \( {H}_{2m} = {H}_{2k} \oplus {H}_{2\left( {m - k}\right) } \) . Since there is an identical isometry\n\n\[ {\operatorname{Id}}_{V} : {H}_{2k} \oplus U \simeq {H}_{2k} \oplus {H}_{2\left( {m - k}\right) } \oplus W, \]\n\nby Lemma 17.2 there exists an isometry \( U \simeq {H}_{2\left( ... | Yes |
Corollary 17.3 Let \( V \) be a space with a nonsingular quadratic form \( q \) and let \( U, W \subset V \) be some subspaces such that both restrictions \( {\left. q\right| }_{U},{\left. q\right| }_{W} \) are nonsingular. Then an isometry \( \varphi : U \simeq W \), if such exists, can be extended (in many ways) to a... | Proof It is enough to show that there exists some isometric isomorphism\n\n\[ \psi : {U}^{ \bot } \simeq {W}^{ \bot } \]\n\nThen \( \varphi \oplus \psi : U \oplus {U}^{ \bot } \simeq W \oplus {W}^{ \bot },\left( {u,{u}^{\prime }}\right) \mapsto \left( {\varphi \left( {h}^{\prime }\right) ,\psi \left( {u}^{\prime }\righ... | Yes |
For each integer \( k \) in the range \( 1 \leq k \leq \dim V/2 \), the orthogonal group of every nonsingular quadratic form on \( V \) acts transitively on \( k \) -dimensional isotropic subspaces and on \( {2k} \) -dimensional hyperbolic subspaces in \( V \) . | The claim about hyperbolic subspaces follows from Corollary 17.3. It implies the claim about isotropic subspaces by Proposition 17.1. | No |
Lemma 17.3 For every \( a, b \in {\mathbb{F}}_{p}^{ * } \) and \( c \in {\mathbb{F}}_{p} \), the equation \( a{x}_{1}^{2} + b{x}_{2}^{2} = c \) is solvable in \( {x}_{1},{x}_{2} \in {\mathbb{F}}_{p} \) . | Proof As \( {x}_{1},{x}_{2} \) run independently through \( {\mathbb{F}}_{p} \), both quantities \( a{x}_{1}^{2} \) and \( c - b{x}_{2}^{2} \) take \( \left( {p + 1}\right) /2 \) different values in \( {\mathbb{F}}_{p} \) . Since \( \left| {\mathbb{F}}_{p}\right| = p \), these two sets of values have at least one commo... | Yes |
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