4. Let A be a square matrix. Assume that Ak = 0 for some positive integer...
Let A be a diagonalizable n × n matrix and let P be an invertible n × n matrix such that B = P−1AP is the diagonal form of A. Prove that Ak = PBkP−1, where k is a positive integer. Use the result above to find the indicated power of A. A = −4 0 4 −3 −1 4 −6 0 6 , A5
An n x n matrix is called nilpotent if Ak = 0 (the zero matrix) for some positive integer k. (a) Suppose A is a nilpotent nxn matrix. Prove that is an eigenvalue of A. (b) Must O be the only eigenvalue of A? Either prove or give a counterexample,
(4) Let A be an n × n matrix for which Ak = 0 for some k > n. Show that An = 0.
Let A be a diagonalizable n x n matrix and let P be an invertible n x n matrix such that B = P-1AP is the diagonal form of A. Prove that Ak = Pekp-1, where k is a positive integer. Use the result above to find the indicated power of A. 0-2 02-2 3 0 -3 ,45 A5 = 11
9. A square matrix A is said to be nilpotent if A 0 for some integer r 21. Let A, B be nilpotent matrices, of the same size, and assume AB BA. Show that AB and A +B are nilpotent 9. A square matrix A is said to be nilpotent if A 0 for some integer r 21. Let A, B be nilpotent matrices, of the same size, and assume AB BA. Show that AB and A +B are nilpotent
Let A be a diagonalizable n x n matrix and let P be an invertible n x n matrix such that B = p-1AP is the diagonal form of A. Prove that Ak = pokp-1, where k is a positive integer. Use the result above to find the indicated power of A. -10 -18 A = 6 11 18].45 -253 -378 A6 = 126 188 11
A matrix A E Mnxn (F) is called nilpotent if, for some positive integer k, Ak O. A" O 1.Show that A eE Mnxn(F) is nilpotent the characteristic polynomial of A is t" 2. Show that if A, BE Mnxn(F) BA, then A + B is nilpotent. nilpotent and AB are 3. Show that if A, B e Mxn(F), A is nilpotent and AB BA, then AB is nilpotent. 4. If A E Mnxn(F) is nilpotent, find the inverse of...
Problem 1 Let {ak} and {bk} be sequences of positive real numbers. Assume that lim “k = 0. k+oo bk 1. Prove that if ) bk converges, so does 'ak k=1 k=1 2. If ) bk diverges, is it necessary that ) ak diverges? k=1 k=1
Prove that, for large integer k 〉 0, the 2-norm of an arbitrary matrix Ak behaves asymptotically like ー2+1 where j is the largest order of all diagonal submatrices J of the Jordan form with o(%)-ρ(A) and v is a positive constant. (Hint: refer to Greenbaum for an expression of the kth power of a j-by-j Jordan block)
Suppose A is a 3 x 3 matrix that is nilpotent but not zero. So Ak-0 for some k 〉 1 A. Verify that 0 is an eigenvalue of A B. Verify that 0 is the only eigenvalue of A. C. Is it possible that there is an invertible matrix P such that P- AP is diagonal?