2. Hermitian matrices. Suppose we are interested in solving an eigenvalue problem with an n by n Hermitian matrix H (H = H where H = (HT)* = (H*)T). H₁ =

A₁₁, i = 1,...,n, (1) where A, is the i-th eigenvalue and ; is the i-th (n dimensional) eigenvector, respectively. Assume all eigenvalues are different for this problem (non-degenerate case). (a) Show that all eigenvalues are real, if H is Hermitian. Note that eigenvectors could be complex in general, while eigenvalues of a Hermitian matrix are real. (1 point) (b) Show that two eigenvectors with two different eigenvalues are orthogonal, i.e., = 0 (or = 0 in matrix notation) for i j. (1 point)