403/503- Homework 4

This set is due Monday, April 11. The questions in problem 2 are required from everybody, and graduate students should also work on problem 1. (Of course, it would make me happier if everybody attempts problem 1 as well.)

1. Let $V$ be a real finite-dimensional, inner product space. For $T\in {\mathcal L}(V)$ , define

$\|T\|_1=\sup_{\|u\|=1}\|Tu\|$ ,

and

$\|T\|_2=\sup_{\|u\|=1}|\langle u,Tu\rangle|$ .

a. Prove that $\|\cdot\|_1$ is a norm on the vector space ${\mathcal L}(V)$ . In particular, $\|T\|_1<\infty$ for all $T$ . Also, prove that $\|Tu\|\le\|T\|_1\|u\|$ for all $u.$
b. Prove that $\|T\|_2\le\|T\|_1$ for all $T$ . Is $\|\cdot\|_2$ also a norm?
c. Prove that for any $T$ there are vectors $u_0,u_1$ of norm 1 with $\|T\|_1=\|Tu_0\|$ and $\|T\|_2=|\langle u_1,Tu_1\rangle|$ .
d. Suppose now that $T$ is such that $\|T\|_1=\|T\|_2$ . Prove (without appealing to the fundamental theorem of algebra and without using determinants) that $T$ admits an eigenvalue $\lambda$ (real) with eigenvector $u_1$ as in item c and, in fact, $\lambda=\langle u_1,Tu_1\rangle$ .
e. Prove that for any $T$ , we have $\|T^2\|_1\le\|T\|_1^2$ .
f. Suppose that $T$ is self-adjoint. Check that so is $T^2$ and that $\|T^2\|_2=\|T^2\|_1$ . In particular, this gives a proof that squares of self-adjoint operators on real vector spaces admit eigenvalues that does not use the fundamental theorem of algebra. Check that the eigenvalues of $T^2$ are non-negative.
g. Again, let $T$ be self-adjoint. (So we know there is an orthonormal basis for $V$ consisting of eigenvectors of $T.$ ) Assume also that $T$ is invertible, that there is a unique eigenvalue $\lambda$ of $T$ of largest absolute value, and that this $\lambda$ satisfies ${\rm dim}({\rm null}(T-\lambda I))=1$ . Let $u$ be an eigenvector of $T$ with eigenvalue $\lambda$ and such that $\|u\|=1$ . Starting with a vector $v_0$ of norm 1 (arbitrary except for the fact that $v_0$ is not orthogonal to $u$ ), define a sequence $v_1,v_2,\dots$ of unit vectors by setting

$v_{k+1}=\displaystyle \frac1{\|Tv_k\|}Tv_k$

(and note we are not dividing by 0, so these vectors are well defined). Also, define a sequence of numbers $\lambda_1,\lambda_2,\dots$ by setting

$\lambda_k=\langle v_k,Tv_k\rangle.$

Prove that there is a sequence $\epsilon_1,\epsilon_2,\dots$ with each $\epsilon_k$ equal to 1 or $-1$ and such that

$\|v_k-\epsilon_k u\|\to 0$

and

$|\lambda_k-\lambda|\to 0$

as $k\to\infty$ .

2. Solve problems 7.1, 7.3, 7.6, 7.7, 7.11, 7.14 from the book.

Note: In problem 1.f, the eigenvalues of $T^2$ are precisely the squares of the eigenvalues of $T$ , but at the moment I do not have a way of showing this directly. As extra-credit, show without appealing to the fundamental theorem of algebra (and without using determinants, of course) that $T$ must have a real eigenvalue.

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