Chapter 13

Interpret the rank of a matrix \(A\) in terms of the dimensions of the image and kernel of the linear map \(L_{A}\).

Let \(V, W\) be finite dimensional vector spaces over a field \(k\). Suppose given non-degenerate bilinear forms on \(V\) and \(W\) respectively, denoted both by \((,\), . Let \(L: V \rightarrow W\) be a surjective linear map and let \(L\) be its transpose; that is, \(\langle L v, w\rangle=\left\langle v,{ }^{\prime} L w\right\rangle\) for \(v \in V\) and \(w \in W\) (a) Show that \({ }^{\prime} L\) is injective. (b) Assume in addition that if \(v \in V, v \neq 0\) then \(\langle v, v\rangle \neq 0 .\) Show that $$ V=\operatorname{Ker} L \oplus \operatorname{Im}^{\prime} L, $$ and that the two summands are orthogonal. (Cf. Exercise 33 for an example.)

Let \(A_{1} \ldots . A\), be row vectors of dimension \(n\), over a field \(k .\) Let \(X=\left(x_{1}, \ldots, x_{n}\right) .\) Let \(b_{1} \ldots \ldots, b, \in k .\) By a system of linear equations in \(k\) one means a system of type $$ A_{1} \cdot X=b_{1}, \ldots, A_{r} \cdot X=b_{r} $$ If \(b_{1}=\cdots=b_{r}=0\), one says the system is homogeneous. We call \(n\) the number of variables, and \(r\) the number of equations. A solution \(X\) of the homogeneous system is called trivial if \(x_{i}=0, i=1, \ldots, n\). (a) Show that a homogeneous system of \(r\) linear equations in' \(n\) unknowns with \(n>r\) always has a non-trivial solution. (b) Let \(L\) be a system of homogeneous linear equations over a field \(k\). Let \(k\) be a subfield of \(k^{\prime}\). If \(L\) has a non-trivial solution in \(k\) ', show that it has a non-trivial solution in \(k\).

Let \(M\) be an \(n \times n\) matrix over a field \(k\). Assume that \(\operatorname{tr}(M X)=0\) for all \(n \times n\) matrices \(X\) in \(k\). Show that \(M=O\).

Let \(S\) be a set of \(n \times n\) matrices over a field \(k\). Show that there exists a column vector \(X \neq 0\) of dimension \(n\) in \(k\), such that \(M X=X\) for all \(M \in S\) if and only if there exists such a vector in some extension field \(k\) ' of \(k\).

Let \(\mathrm{H}\) be the division ring over the reals generated by elements \(i, j, k\) such that \(i^{2}=j^{2}=k^{2}=-1\), and $$ i j=-j i=k, \quad j k=-k j=i, \quad k i=-i k=j $$ Then \(\mathrm{H}\) has an automorphism of order 2 , given by $$ a_{0}+a_{1} i+a_{2} j+a_{3} k \mapsto a_{0}-a_{1} i-a_{2} j-a_{3} k $$ Denote this automorphism by \(\alpha \mapsto \bar{\alpha}\). What is \(\alpha \bar{x}\) ? Show that the theory of hermitian forms can be carried out over \(\mathbf{H}\), which is called the division ring of quaternions (or by abuse of language, the non-commutative field of quaternions).

Let \(N\) be a strictly upper truangular \(n \times n\) matrix, that is \(N=\left(a_{i j}\right)\) and \(a_{i j}=0\) if \(i \geqq j\). Show that \(N^{n}=0\).

Let \(E\) be a vector space over \(k\), of dimension \(n\). Let \(T: E \rightarrow E\) be a linear map such that \(T\) is nilpotent, that is \(T^{m}=0\) for some positive integer \(m\). Show that there exists a basis of \(E\) over \(k\) such that the matrix of \(T\) with respect to this basis is strictly upper triangular.

If \(N\) is a nilpotent \(n \times n\) matrix, show that \(I+N\) is invertible.

Let \(R\) be the set of all upper triangular \(n \times n\) matrices \(\left(a_{i j}\right)\) with \(a_{i j}\) in some field \(k\), so \(a_{i j}=0\) if \(i>j .\) Let \(J\) be the set of all strictly upper triangular matrices. Show that \(J\) is a two- sided ideal in \(R\). How would you describe the factor ring \(R / \underline{J} ?\)

Let \(G\) be the group of upper triangular matrices with non-zero diagonal elements. Let \(H\) be the subgroup consisting of those matrices whose diagonal element is 1 . (Actually prove that \(H\) is a subgroup). How would you describe the factor group \(G / H ?\)

Let \(R\) be the ring of \(n \times n\) matrices over a field \(k .\) Let \(L\) be the subset of matrices which are 0 except on the first column. (a) Show that \(L\) is a left ideal. (b) Show that \(L\) is a minimal left ideal; that is, if \(L^{\prime} \subset L\) is a left ideal and \(L^{\prime} \neq 0\), then \(L^{\prime}=L .\) (For more on this situation, see Chapter VII, \(\left.85 .\right)\)

Let \(F\) be any field. Let \(D\) be the subgroup of diagonal matrices in \(G L_{n}(F)\). Let \(N\) be the normalizer of \(D\) in \(G L_{n}(F)\). Show that \(N / D\) is isomorphic to the symmetric group on \(n\) clements.

Let \(F\) be a finite field with \(q\) elements. Show that the order of \(G L_{n}(F)\) is $$ \left(q^{n}-1\right)\left(q^{n}-q\right) \cdots\left(q^{n}-q^{n-1}\right)=q^{m i n-11 / 2} \prod_{i=1}^{n}\left(q^{i}-1\right) . $$

Again let \(F\) be a finite field with \(q\) elements. Show that the order of \(S L_{n}(F)\) is $$ q^{n s-1 M 2} \prod_{i=2}^{n}\left(q^{i}-1\right) $$ and that the order of \(P S L_{m}(F)\) is $$ \frac{1}{d} q^{n i n-1 \sqrt 2} \prod_{i=2}^{n-1}\left(q^{i}-1\right) $$ where \(d\) is the greatest common divisor of \(n\) and \(q-1\).

Let \(F\) be a finite field with \(q\) elements. Show that the group of all upper trangular matrices with 1 on the diagonal is a Sylow subgroup of \(G L_{n}(F)\) and of \(S L_{n}(F)\).

Show that the order of \(S L_{2}(\mathbf{Z} / N Z)\) is equal to $$ N^{3} \prod_{p \mid N}\left(1-\frac{1}{p^{2}}\right), $$ where the product is taken over all primes dividing \(N\).

Show that one has an exact sequence $$ 1 \rightarrow S L_{2}(\mathbf{Z} / N \mathbf{Z}) \rightarrow G L_{2}(\mathbf{Z} / N \mathbf{Z}) \stackrel{d e !}{\leftrightarrow}(\mathbf{Z} / N \mathbf{Z})^{*} \rightarrow 1 . $$ In fact, show that $$ G L_{2}(\mathbf{Z} / \mathrm{NZ})=S L_{2}(\mathbf{Z} / N Z) G_{N} $$ where \(G_{N}\) is the group of matrices $$ \left(\begin{array}{ll} 1 & 0 \\ 0 & d \end{array}\right) \text { with } d \in(\mathbf{Z} / N Z)^{\bullet} $$

Show that \(S L_{2}(\mathbf{Z})\) is generated by the matrices $$ \left(\begin{array}{ll} 1 & 1 \\ 0 & 1 \end{array}\right) \text { and }\left(\begin{array}{rr} 0 & -1 \\ 1 & 0 \end{array}\right) . $$

Let \(p\) be a prime \(\geqq 5\). Let \(G\) be a subgroup of \(S L_{2}\left(\mathbf{Z} / p^{n} \mathbf{Z}\right)\) with \(n \geq 1\). Assume that the image of \(G\) in \(S L_{2}(\mathbf{Z} / p \mathbf{Z})\) under the natural homomorphism is all of \(S L_{2}(\mathbf{Z} / p \mathbf{Z})\). Prove that \(G=S L_{2}\left(\mathbf{Z} / p^{n} Z\right)\).

Let \(k\) be a field in which every quadratic polynomial has a root. Let \(B\) be the Borel subgroup of \(G L_{2}(k) .\) Show that \(G\) is the union of all the conjugates of \(B\). (This cannot happen for finite groups!)

Let \(A, B\) be square matrices of the same size over a field \(k\). Assume that \(B\) is nonsingular. If \(t\) is a variable, show that \(\operatorname{det}(A+t B)\) is a polynomial in \(t\), whose leading coefficient is \(\operatorname{det}(B)\), and whose constant term is \(\operatorname{det}(A)\)

Let \(A\) be a commutative ring, and \(I=\left(x_{1} \ldots \ldots x_{r}\right)\) an ideal. Let \(c_{i j} \in A\) and let $$ y_{i}=\sum_{j=1}^{\prime} c_{i j} x_{j} $$ Let \(l^{\prime}=\left(y_{1} \ldots \ldots y_{f}\right)\). Let \(D=\operatorname{det}\left(c_{i j}\right)\). Show that \(D I \subset I\) :

Let \(G\) be a finite commutative group and let \(F\) be the vector space of functions of \(G\) into \(\mathbf{C}\). Show that the characters of \(G\) (homomorphisms of \(G\) into the roots of unity) form a basis for this space. If \(f: G \rightarrow \mathbf{C}\) is a function, show that for \(a, b \in G\). $$ \operatorname{det}\left(f\left(a b^{-1}\right)\right)=\prod_{x} \sum_{a \in G} \chi(a) f(a) $$ where the product is taken over all characters. [Hint: Use both the characters and the characteristic functions of elements of \(G\) as bases for \(F\), and consider the linear map $$ T=\sum f(a) T_{a} $$ where \(T_{a}\) is translation by \(\left.a .\right]\) Also show that $$ \operatorname{det}\left(f\left(a b^{-1}\right)\right)=\left(\sum_{a \in G} f(a)\right) \operatorname{det}\left(f\left(a b^{-1}\right)-f\left(b^{-1}\right)\right) $$ where the determinant on the left is taken for all \(a, b \in G\), and the determinant on the right is taken only for \(a, b \neq 1\).

Prove that \(H^{1}\left(G, G L_{0}(K)\right)=1\). [Hint: Let \(\left\\{e_{1}, \ldots, e_{N}\right\\}\) be a basis of Mat \(_{n}(k)\) over \(k\), say the matrices with 1 in some component and 0 elsewhere. Let $$ x=\sum_{i=1}^{N} x_{i} e_{i} $$ with variables \(x_{i}\). There exists a polynomial \(P(X)\) such that \(x\) is invertible if and only if \(P_{1}\left(x_{1}, \ldots, x_{N}\right) \neq 0 .\) Instead of \(P\left(x_{1}, \ldots, x_{N}\right)\) we also write \(P(x)\). Let \(\\{A(\sigma)\\}\) be a cocycle. Let \(\left\\{t_{a}\right\\}\) be algebraically independent variables over \(k\). Then $$ P\left(\sum_{\gamma \in G} t_{y} A(y)\right) \neq 0 $$ because the polynomial does not vanish when one \(t_{y}\) is replaced by 1 and the others are replaced by 0. By the algebraic independence of automorphisms from Galois theory, there exists an element \(y \in K\) such that if we put $$ B=\sum_{\eta}(\gamma y) A(\gamma) $$ then \(P(B) \neq 0\), so \(B\) is invertible. It is then immediately verified that \(A(\sigma)=B \sigma B^{-1}\). But when \(k\) is finite, cf. my Algebraic Groups over Finite Fields, Am. J. Vol 78 No. 3. \(1956 .]\)

(Kolchin-Lang, Proc. AMS Vol 11 No. 1,1960 ). Let \(K\) be a finite Galois extension of \(k, G=\operatorname{Gal}(K / k)\) as in the preceding exercise. Let \(V\) be a finite-dimensional vector space over \(K\), and suppose \(G\) operates on \(V\) in such a way that \(\sigma(a v)=\sigma(a) \sigma(v)\) for \(a \in K\) and \(v \in V\). Prove that there exists a basis \(\left\\{w_{1}, \ldots, w_{n}\right\\}\) such that \(\sigma w_{i}=w_{i}\) for all \(i=1, \ldots, n\) and all \(\sigma \in G\) (an invariant basis). Hint: Let \(\left\\{v_{1}, \ldots, v_{n}\right\\}\) be any basis, and let $$ \sigma\left(\begin{array}{c} v_{1} \\ \vdots \\ v_{n} \end{array}\right)=A(\sigma)\left(\begin{array}{c} v_{1} \\ \vdots \\ v_{\alpha} \end{array}\right) $$ where \(A(\sigma)\) is a matrix in \(G L_{n}(K)\). Solve for \(B\) in the equation \((\sigma B) A(\sigma)=B\), and let $$ \left(\begin{array}{c} w_{1} \\ \vdots \\ w_{n} \end{array}\right)=B\left(\begin{array}{c} v_{1} \\ \vdots \\ v_{n} \end{array}\right) $$

(Continuation of Exercise 34). Prove that the representation of \(O(n)=U_{n}(\mathbf{R})\) on \(\operatorname{Har}(n, d)\) is irreducible. Readers will find a proof in the following: The Howe-Tan proof runs as follows. We now use the hermitian product $$ \langle P, Q\rangle=\int_{\mathbf{s}^{n-1}} P(x) \overline{Q(x)} d \sigma(x) $$ where \(\sigma\) is the rotation invariant measure on the \((n-1)\) -sphere \(\mathbf{S}^{n-1}\). Let \(e_{1}, \ldots, e_{n}\) be the unit vectors in \(\mathbf{R}^{n} .\) We can identify \(O(n-1)\) as the subgroup of \(O(n)\) leaving \(e_{n}\) fixed. Observe that \(O(n)\) operates on \(\operatorname{Har}(n, d)\), say on the right by composition \(P \mapsto P \circ A, A \in O(n)\), and this operation commutes with \(\Delta\). Let $$ \lambda: \operatorname{Har}(n, d) \rightarrow \mathbf{C} $$ be the functional such that \(\lambda(P)=P\left(e_{n}\right)\). Then \(\lambda\) is \(O(n-1)\) -invariant, and since the hermitian product is non-degenerate, there exists a harmonic polynomial \(Q_{n}\) such that $$ \lambda(P)=\left\langle P, Q_{n}\right\rangle \quad \text { for all } P \in \operatorname{Har}(n, d) . $$ Let \(M \subset \operatorname{Har}(n, d)\) be an \(O(n)\) -submodule. Then the restriction \(\lambda_{M}\) of \(\lambda\) to \(M\) is nontrivial because \(O(n)\) acts transitively on \(\mathbf{S}^{n-1}\). Let \(Q_{n}^{M}\) be the orthogonal projection of \(Q_{n}\) on \(M\). Then \(Q_{n}^{M}\) is \(O(n-1)\) -invariant, and so is a linear combination $$ Q_{n}^{M}(x)=\sum_{j+2 k=d} c_{j} x_{n}^{j} r_{n-1}^{2 k} $$ Furthermore \(Q_{n}^{H}\) is harmonic. From this you can show that \(Q_{n}^{H}\) is uniquely determined, by showing the existence of recursive relations among the coefficients \(c_{j}\). Thus the submodule \(M\) is uniquely determined, and must be all of \(\operatorname{Har}(n, d)\).

Let \(F\) be a field of characteristic \(0 .\) Let \(g=5 l_{n}(F)\) be the vector space of matrices with trace 0 , with its Lie algebra structure \([X, Y]=X Y-Y X\). Let \(E_{i j}\) be the matrix having \((i, j)\) -component 1 and all other components \(0 .\) Let \(G=S L_{n}(F) .\) Let \(A\) be the multiplicative group of diagonal matrices over \(F\). (a) Let \(H_{i}=E_{i i}-E_{i+1, i+1}\) for \(i=1, \ldots, n-1\). Show that the elements \(E_{j}\) \((i \neq j), H_{1}, \ldots, H_{n-1}\) form a basis of \(g\) over \(F .\) (b) For \(g \in G\) let \(\mathbf{c}(g)\) be the conjugation action on \(g\). that is \(\mathbf{c}(g) X=g X g^{-1}\). Show that each \(E_{j}\) is an cigenvector for this action restricted to the group \(A\). (c) Show that the conjugation representation of \(G\) on \(g\) is irreducible, that is, if \(V \neq 0\) is a subspace of \(\mathrm{g}\) which is \(\mathrm{c}(G)\) -stable, then \(V=\mathrm{g} .\) Hint: Look up the sketch of the proof in [JoL 01], Chapter VII, Theorem \(1.5\), and put in all the details. Note that for \(i \neq j\) the matrix \(E_{y}\) is n?lpotent, so for variable \(t\). the exponential series \(\exp \left(t E_{y}\right)\) is actually a polynomial. The derivative with respect to \(t\) can be taken in the formal power series \(F[[t]]\), not using limits. If \(X\) is a matrix, and \(x(t)=\exp (t X)\), show that $$ \left.\left.\frac{d}{d t} x(t) Y x(t)^{-1}\right|_{t=0}=X Y-Y X=\mid X, Y\right] $$