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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

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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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