Chapter 15

(a) Let \(E\) be a finite dimensional space over the complex numbers, and let $$ h: E \times E \rightarrow \mathbf{C} $$ be a hermitian form. Write $$ h(x, y)=g(x, y)+i f(x, y) $$ where g. \(f\) are real valued. Show that \(g . f\) are \(\mathbf{R}\) -bilinear, \(g\) is symmetric, \(f\) is alternating. (b) Let \(E\) be finite dimensional over \(\mathbf{C}\). Let \(g: E \times E \rightarrow \mathbf{C}\) be \(\mathbf{R}\) -bilinear. Assume that for all \(x \in E\), the map \(y \mapsto g(x, y)\) is C-linear, and that the \(R\) -bilinear form $$ f(x, y)=g(x, y)-g(y, x) $$ is real-valued on \(E \times E\). Show that there exists a hermitian form \(h\) on \(E\) and \(\mathrm{A}\) symmetric \(\mathbf{C}\) -bilinear form \(\psi\) on \(E\) such that \(2 i g=h+\psi\). Show that \(h\) and \(\psi\) are uniquely determined.

Prove the real case of the unitary spectral theorem: If \(E\) is a non-zero finite dimensional space over \(\mathbf{R}\), with a positive definite symmetric form, and \(U: E \rightarrow E\) is a unitary linear map, then \(E\) has an orthogonal decomposition into subspaces of dimension 1 or 2 . invariant under \(U\). If \(\operatorname{dim} E=2\), then the matrix of \(U\) with respect to any orthonormal basis is of the form $$ \left(\begin{array}{rr} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right) \text { or }\left(\begin{array}{rr} -1 & 0 \\ 0 & 1 \end{array}\right)\left(\begin{array}{cc} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right) $$ depending on whether \(\operatorname{det}(U)=1\) or \(-1 .\) Thus \(U\) is a rotation, or a rotation followed by a reflection.

Let \(E\) be a finite-dimensional, non-zero vector space over the reals, with a positive definite scalar product. Let \(T: E \rightarrow E\) be a unitary automorphism of \(E .\) Show that \(E\) is an orthogonal sum of subspaces $$ E=E_{1} \perp \cdots \perp E_{m} $$ such that each \(E_{i}\) is \(T\) -invariant, and has dimension 1 or \(2 .\) If \(E\) has dimension 2, show that one can find a basis such that the matrix associated with \(T\) with respect to this basis is $$ \left(\begin{array}{rr} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right) \quad \text { or } \quad\left(\begin{array}{cc} -\cos \theta & \sin \theta \\ \sin \theta & \cos \theta \end{array}\right) $$ according as det \(T=1\) or det \(T=-1\).

Let \(E\) be a finite dimensional non-zero vector space over \(\mathbf{C}\), with a positive definite hermitian product. Let \(A, B: E \rightarrow E\) be a hermitian endomorphism. Assume that \(A B=B A\). Prove that there exists a basis of \(E\) consisting of common eigenvectors for \(A\) and \(B\).

Let \(E\) be a finite-dimensional space over the complex, with a positive definite hermitian form. Let \(S\) be a set of (C-linear) endomorphisms of \(E\) having no invariant subspace except 0 and \(E\). (This means that if \(F\) is a subspace of \(E\) and \(B F \subset F\) for all \(B \in S\), then \(F=0\) or \(F=E\).) Let \(A\) be a hermitian map of \(E\) into itself such that \(A B=B A\) for all \(B \in S\). Show that \(A=\lambda I\) for some real number \(\lambda\). [Hint: Show that there exists exactly one eigenvalue of \(A\). If there were two cigenvalues, say \(\lambda_{1} \neq \lambda_{2}\), one could find two polynomials \(f\) and \(g\) with real coefficients such that \(f(A) \neq 0, g(A) \neq 0\) but \(f(A) g(A)=0 .\) Let \(F\) be the kernel of \(g(A)\) and get a contradiction.]

(a) Show that \(A \geq 0\) if and only if all eigenvalues of \(A\) belonging to non-zero eigenvectors are \(\geq 0 .\) Both in the hermitian case and the symmetric case, one says that \(A\) is semipositive if \(A \geq 0\), and positive definite if \(\langle A x, x\rangle>0\) for all \(x \neq 0\) (b) Show that an automorphism \(A\) of \(E\) can be written in a unique way as a product \(A=U P\) where \(U\) is real unitary (that is, \({ }^{\prime} U U=I\), and \(P\) is symmetric positive definite. For two hermitian or symmetric endomorphisms \(A, B\), define \(A \geqq B\) to mean \(A-B \geqq 0\), and similarly for \(A>B .\) Suppose \(A>0 .\) Show that there are two real numbers \(\alpha>0\) and \(\beta>0\) such that \(\alpha l \leq A \leq \beta l\).

(a) Show that \(A \geq 0\) if and only if all eigenvalues of \(A\) belonging to non-zero eigenvectors are \(\geq 0 .\) Both in the hermitian case and the symmetric case, one says that \(A\) is semipositive if \(A \geq 0\), and positive definite if \(\langle A x, x\rangle>0\) for all \(x \neq 0\) (b) Show that an automorphism \(A\) of \(E\) can be written in a unique way as a product \(A=U P\) where \(U\) is real unitary (that is, \({ }^{\prime} U U=I\), and \(P\) is symmetric positive definite. For two hermitian or symmetric endomorphisms \(A, B\), define \(A \geqq B\) to mean \(A-B \geqq 0\), and similarly for \(A>B .\) Suppose \(A>0 .\) Show that there are two real numbers \(\alpha>0\) and \(\beta>0\) such that \(\alpha l \leq A \leq \beta l\).

Again, let \(E\) be non-zero finite dimensional over \(\mathbf{R}\), and with a positive definite symmetric form. Let \(A: E \rightarrow E\) be a linear map. Prove: (a) If \(A\) is symmetric (resp. altemating), then \(\exp (A)\) is symmetric positive definite (resp. real unitary). (b) If \(A\) is a linear automorphism of \(E\) sufficiently close to \(I\), and is symmetric

Let \(R\) be a commutative ring, let \(E, F\) be \(R\) -modules, and let \(f: E \rightarrow F\) be a mapping. Assume that multiplication by 2 in \(F\) is an invertible map. Show that \(f\) is homogeneous quadratic if and only if \(f\) satisfies the parallelogram law: $$ f(x+y)+f(x-y)=2 f(x)+2 f(y) $$ for all \(x, y \in E\).

(Tate) Let \(S\) be a set and \(f: S \rightarrow S\) a map of \(S\) into itself. Let \(h: S \rightarrow \mathbf{R}\) be a real valued function. Assume that there exists a real number \(d>1\) such that \(h \circ f-d f\) is bounded. Show that there exists a unique function \(h_{f}\) such that \(h_{f}-h\) is bounded, and \(h_{f} \circ f=d h_{f} .\left[\right.\) Hint: Let \(h_{f}(x)=\lim h\left(f^{\prime \prime}(x)\right) / d^{n}\) "]

Let \(E\) be a vector space over a field \(k\) and let \(g\) be a bilinear form on \(E\). Assume that whenever \(x, y \in E\) are such that \(g(x, y)=0\), then \(g(y, x)=0 .\) Show that \(g\) is symmetric or alternating.

Let \(E\) be a module over \(Z\). Assume that \(E\) is free, of dimension \(n \geq 1\), and let \(f\) be a bilinear alternating form on \(E\). Show that there exists a basis \(\left\\{e_{i}\right\\}(i=1, \ldots, n)\) and an integer \(r\) such that \(2 r \leqq n\), $$ e_{t} \cdot e_{2}=a_{1}, \quad e_{3}+e_{4}=a_{2}, \ldots, e_{2 r-1} \cdot e_{2 r}=a $$ where \(a_{1}, \ldots, a, \in \mathbf{Z}, a_{i} \neq 0\), and \(a_{i}\) divides \(a_{i+1}\) for \(i=1, \ldots, r-1\) and finally \(e_{i} \cdot e_{j}=0\) for all other pairs of indices \(i \leqq j .\) Show that the ideals \(\mathrm{Za}_{i}\) are uniquely determined. [Hint: Consider the injective homomorphism \(\varphi_{f}: E \rightarrow E^{\vee}\) of \(E\) into the

Show that the pfaffian of an alternating \(n \times n\) matrix is 0 when \(n\) is odd.

Let a be the \(\mathbf{R}\) -vector space of real diagonal matrices with trace \(0 .\) Let \(a^{\vee}\) be the dual space. Let \(\alpha_{i}(i=1, \ldots, n-1)\) be the functional defined on an element \(H=\) \(\operatorname{diag}\left(h_{1}, \ldots, h_{k}\right)\) by \(\alpha_{i}(H)=h_{i}-h_{i+1} .\) (a) Show that \(\left\\{\alpha_{1}, \ldots, \alpha_{n-1}\right\\}\) is a basis of \(a^{\vee}\) over \(\mathbf{R}\). (b) Let \(H_{i, i+1}\) be the diagonal matrix with \(h_{i}=1, h_{i+1}=-1\), and \(h_{j}=0\) for \(j \neq i, i+1 .\) Show that \(\left\\{H_{1,2}, \ldots, H_{n-1, n}\right\\}\) is a basis of \(a\). (c) Abbreviate \(H_{i, i+1}=H_{i}(i=1, \ldots, n-1)\). Let \(\alpha_{i}^{\prime} \in a^{\vee}\) be the functional such that \(\alpha_{i}^{\prime}\left(H_{j}\right)=\delta_{i j}\) \(\left(=1\right.\) if \(i=j\) and 0 otherwise). Thus \(\left\\{\alpha_{1}^{\prime}, \ldots, \alpha_{n-1}^{\prime}\right\\}\) is the dual basis of \(\left\\{H_{1}, \ldots, H_{n-1}\right\\}\). Show that $$ \alpha_{i}^{\prime}(H)=h_{1}+\cdots+h_{i} $$

Positivity. On a (real diagonal matrices with trace 0) the form of Exercise 27 can be defined by \(\operatorname{tr}(X Y)\), since elements \(X, Y \in a\) are symmetric. Let \(\mathscr{A}=\left\\{\alpha_{1}, \ldots, \alpha_{n-1}\right\\}\) denote the basis of Exercise 26. Define an element \(H \in \mathrm{a}\) to be semipositive (writen \(H \geqq 0\) ) if \(\alpha_{i}(H) \geqq 0\) for all \(i=1, \ldots, n-1 .\) For each \(\alpha \in a^{\nu}\), let \(H_{a}\) e a represent \(\alpha\) with respect to \(B_{t}\), that is \(\left\langle H_{\alpha}, H\right\rangle=\alpha(H)\) for all \(H \in \mathrm{a} .\) Show that \(H \geqq 0\) if and only if $$ H=\sum_{i=1}^{n-1} s_{i} H_{x_{i}^{\prime}} \quad \text { with } s_{i} \geqq 0 . $$ Similarly, define \(H\) to be positive and formulate the similar condition with \(s_{i}>0 .\)