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 be a real finite-dimensional, inner product space. For , define
a. Prove that is a norm on the vector space . In particular, for all . Also, prove that for all
b. Prove that for all . Is also a norm?
c. Prove that for any there are vectors of norm 1 with and .
d. Suppose now that is such that . Prove (without appealing to the fundamental theorem of algebra and without using determinants) that admits an eigenvalue (real) with eigenvector as in item c and, in fact, .
e. Prove that for any , we have .
f. Suppose that is self-adjoint. Check that so is and that . 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 are non-negative.
g. Again, let be self-adjoint. (So we know there is an orthonormal basis for consisting of eigenvectors of ) Assume also that is invertible, that there is a unique eigenvalue of of largest absolute value, and that this satisfies . Let be an eigenvector of with eigenvalue and such that . Starting with a vector of norm 1 (arbitrary except for the fact that is not orthogonal to ), define a sequence of unit vectors by setting
(and note we are not dividing by 0, so these vectors are well defined). Also, define a sequence of numbers by setting
Prove that there is a sequence with each equal to 1 or and such that
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 are precisely the squares of the eigenvalues of , 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 must have a real eigenvalue.