15. Let be an eigenvalue of the n n matrix A and x = 0 be an associated eigenvector. a
Chapter 7, Problem 15(choose chapter or problem)
15. Let be an eigenvalue of the n n matrix A and x = 0 be an associated eigenvector. a. Show that is also an eigenvalue of At . b. Show that for any integer k 1, k is an eigenvalue of Ak with eigenvector x. c. Show that if A1 exists, then 1/ is an eigenvalue of A1 with eigenvector x. d. Generalize parts (b) and (c) to (A1)k for integers k 2. e. Given the polynomial q(x) = q0 + q1x + + qk xk , define q(A) to be the matrix q(A) = q0I + q1A ++ qkAk . Show that q() is an eigenvalue of q(A) with eigenvector x. f. Let = be given. Show that if A I is nonsingular, then 1/( ) is an eigenvalue of (A I)1 with eigenvector x.
Unfortunately, we don't have that question answered yet. But you can get it answered in just 5 hours by Logging in or Becoming a subscriber.
Becoming a subscriber
Or look for another answer