Let A be an n n matrix with distinct real eigenvalues 1,2 . . . .

Let A be an n × n matrix with distinct real eigenvalues λ1,λ2 . . . . λn. Let λ be a scalar that is not an eigenvalue of A and let B = (A - λI)-1. Show that
(a) The scalars μj = l/(λj - λ), j = 1,... ,n are the eigenvalues of B.
(b) If xJ is an eigenvector of B belonging to μj, then xj is an eigenvector of A belonging to λj.
(c) Show that if the power method is applied to B. then the sequence of vectors will converge to an eigenvector of A belonging to the eigenvalue that is closest to λ. [The convergence will be rapid if λ is much closer to one A., than to any of the others. This method of computing eigenvectors using powers of (A - λI)-1 is called the inverse power method.]