Let be an Hermitian matrix. Show that all the eigenvalues of are real.
Suppose now that has distinct eigenvalues.
(a) Show that the eigenvectors of are orthogonal.
(b) Define the characteristic polynomial of . Let
Prove the matrix identity
(c) What is the range of possible values of
for non-zero vectors Justify your answer.
(d) For any (not necessarily symmetric) real matrix with real eigenvalues, let denote its maximum eigenvalue. Is it possible to find a constant such that
for all non-zero vectors and all such matrices ? Justify your answer.