The Inverse Power Method. Let A be a nonsingular matrix. (a) Show that the eigenvalues of A-1

Question:

The Inverse Power Method. Let A be a nonsingular matrix.
(a) Show that the eigenvalues of A-1 are the reciprocals 1/λ of the eigenvalues of A. How are the eigenvectors related?
(b) Show how to use the power method on A-1 to produce the smallest (in modulus) eigenvalue of A.
(c) What is the rate of convergence of the algorithm?
(d) Design a practical iterative algorithm based on the (permuted) LU decomposition of A.