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Problem 2 Suppose A is an n x n matrix with eigenvalue A and corresponding eigenvector x. (a) Prove that for any complex scalar c, the matrix cA has eigenvalue ca with cor- responding eigenvector x. (b) Prove that for any positive integer r, the matrix A' has eigenvalue A' with cor- responding eigenvector x. ' (c) Let p(x) = co+cjx+c2x² + • ·+ C;x* be a polynomial function. Prove that p(1) is an eigenvalue of p(A) with corresponding eigenvector x. %3D

Problem 2 Suppose A is an n x n matrix with eigenvalue A and corresponding eigenvector x. (a) Prove that for any complex scalar c, the matrix cA has eigenvalue ca with cor- responding eigenvector x. (b) Prove that for any positive integer r, the matrix A' has eigenvalue A' with cor- responding eigenvector x. ' (c) Let p(x) = co+cjx+c2x² + • ·+ C;x* be a polynomial function. Prove that p(1) is an eigenvalue of p(A) with corresponding eigenvector x. %3D

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter5: Orthogonality
Section5.3: The Gram-schmidt Process And The Qr Factorization
Problem 1BEXP
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Problem 2 Suppose A is an n x n matrix with eigenvalue A and corresponding eigenvector x. (a) Prove that for any complex scalar c, the matrix cA has eigenvalue cd with cor- responding eigenvector x. (b) Prove that for any positive integer r, the matrix A has eigenvalue l with cor- responding eigenvector x. ' (c) Let p(x) = co+c1x+c2x² + • · + C;x* be a polynomial function. Prove that p(1) is an eigenvalue of p(A) with corresponding eigenvector x.
Transcribed Image Text:Problem 2 Suppose A is an n x n matrix with eigenvalue A and corresponding eigenvector x. (a) Prove that for any complex scalar c, the matrix cA has eigenvalue cd with cor- responding eigenvector x. (b) Prove that for any positive integer r, the matrix A has eigenvalue l with cor- responding eigenvector x. ' (c) Let p(x) = co+c1x+c2x² + • · + C;x* be a polynomial function. Prove that p(1) is an eigenvalue of p(A) with corresponding eigenvector x.
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