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\|_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


|\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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