Consider the n x n symmetric matrix H = I - 2uuT, where u is a unit vector in Rn, i.e., uTu = 1. (a) Show that H is an orthogonal matrix. (b) Show that u is an eigenvector of H and find the corresponding eigenvalue.

Consider the matrix 2 2 A = 4 2 2 (a) Compute the eigenvalues and eigenvectors of A. (b) Find rank one matrices B and C such that A = B + C and BC 0, explaining why the matrices you found have these properties. (hint: use your answer to part (a))

Let A be a symmetric n x n matrix. Let A1 and A2 be two eigenvalues with eigenvectors vị and v2 for A, respectively. Show that vi and v2 are orthogonal.

Consider the matrix A = [2 0           3 4]. Show that 2 and 4 are eigenvalues of A and find all corresponding eigen- vectors. Find an eigenbasis for A and thus diagonalize A.