91Ó°ÊÓ

Prove the following results. (a) Any square diagonal matrix is symmetric. (b) Any matrix congruent to a diagonal matrix is symmetric. (c) the corollary to Theorem \(6.35\)

Let \(V\) be a finite-dimensional inner product space over \(F\). (a) Parseval's Identity. Let \(\left\\{v_{1}, v_{2}, \ldots, v_{n}\right\\}\) be an orthonormal basis for \(\mathrm{V}\). For any \(x, y \in \mathrm{V}\) prove that $$ \langle x, y\rangle=\sum_{i=1}^{n}\left\langle x, v_{i}\right\rangle \overline{\left\langle y, v_{i}\right\rangle} . $$ (b) Use (a) to prove that if \(\beta\) is an orthonormal basis for \(V\) with inner product \(\langle\cdot, \cdot\rangle\), then for any \(x, y \in \mathrm{V}\) $$ \left\langle\phi_{\beta}(x), \phi_{\beta}(y)\right\rangle^{\prime}=\left\langle[x]_{\beta},[y]_{\beta}\right\rangle^{\prime}=\langle x, y\rangle, $$ where \(\langle\cdot, \cdot\rangle^{\prime}\) is the standard inner product on \(\mathrm{F}^{n}\).

(a) Prove that if \(\mathrm{V}\) is an inner product space, then \(|\langle x, y\rangle|=\|x\| \cdot\|y\|\) if and only if one of the vectors \(x\) or \(y\) is a multiple of the other. Hint: If the identity holds and \(y \neq 0\), let $$ a=\frac{\langle x, y\rangle}{\|y\|^{2}}, $$ and let \(z=x-a y\). Prove that \(y\) and \(z\) are orthogonal and $$ |a|=\frac{\|x\|}{\|y\|} . $$ Then apply Exercise 10 to \(\|x\|^{2}=\|a y+z\|^{2}\) to obtain \(\|z\|=0\). (b) Derive a similar result for the equality \(\|x+y\|=\|x\|+\|y\|\), and generalize it to the case of \(n\) vectors.

Let \(U\) be a unitary operator on an inner product space \(V\), and let \(W\) be a finite-dimensional U-invariant subspace of V. Prove that (a) \(U(W)=W\); (b) \(\mathrm{W}^{\perp}\) is U-invariant. Contrast (b) with Exercise 16 .

(a) Bessel's Inequality. Let \(\mathrm{V}\) be an inner product space, and let \(S=\) \(\left\\{v_{1}, v_{2}, \ldots, v_{n}\right\\}\) be an orthonormal subset of \(\mathrm{V}\). Prove that for any \(x \in \mathrm{V}\) we have $$ \|x\|^{2} \geq \sum_{i=1}^{n}\left|\left\langle x, v_{i}\right\rangle\right|^{2} . $$ Hint: Apply Theorem \(6.6\) to \(x \in \mathrm{V}\) and \(\mathrm{W}=\operatorname{span}(S)\). Then use Exercise 10 of Section 6.1. (b) In the context of (a), prove that Bessel's inequality is an equality if and only if \(x \in \operatorname{span}(S)\).

Prove that a matrix that is both unitary and upper triangular must be a diagonal matrix.

Let \(\mathrm{T}\) and \(\mathrm{U}\) be self-adjoint linear operators on an \(n\)-dimensional inner product space \(\mathrm{V}\), and let \(A=[\mathrm{T}]_{\beta}\), where \(\beta\) is an orthonormal basis for V. Prove the following results. (a) \(\mathrm{T}\) is positive definite [semidefinite] if and only if all of its eigenvalues are positive [nonnegative]. (b) \(\mathrm{T}\) is positive definite if and only if $$ \sum_{i, j} A_{i j} a_{j} \bar{a}_{i}>0 \text { for all nonzero } n \text {-tuples }\left(a_{1}, a_{2}, \ldots, a_{n}\right) \text {. } $$ (c) \(\mathrm{T}\) is positive semidefinite if and only if \(A=B^{*} B\) for some square matrix \(B\). (d) If \(T\) and \(U\) are positive semidefinite operators such that \(T^{2}=U^{2}\), then \(\mathrm{T}=\mathrm{U}\). (e) If \(T\) and \(U\) are positive definite operators such that \(T U=U T\), then TU is positive definite. (f) \(\mathrm{T}\) is positive definite [semidefinite] if and only if \(A\) is positive definite [semidefinite]. Because of (f), results analogous to items (a) through (d) hold for matrices as well as operators.

Let \(\mathrm{V}\) be a vector space over \(F\), where \(F=R\) or \(F=C\), and let \(\mathrm{W}\) be an inner product space over \(F\) with inner product \(\langle\cdot, \cdot\rangle .\) If \(\mathrm{T}: \mathrm{V} \rightarrow \mathrm{W}\) is linear, prove that \(\langle x, y\rangle^{\prime}=\langle\mathrm{T}(x), \mathrm{T}(y)\rangle\) defines an inner product on \(\mathrm{V}\) if and only if \(\mathrm{T}\) is one-to-one.

Let \(A\) be an \(n \times n\) matrix. Prove that \(\operatorname{det}\left(A^{*}\right)=\overline{\operatorname{det}(A)}\). Visit goo.gl/csqoFY for a solution.

Let \(\mathrm{T}\) and \(\mathrm{U}\) be positive definite operators on an inner product space V. Prove the following results. (a) \(\mathrm{T}+\mathrm{U}\) is positive definite. (b) If \(c>0\), then \(c \mathrm{~T}\) is positive definite. (c) \(\mathrm{T}^{-1}\) is positive definite. Visit goo.gl/cQch7i for a solution.