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7. Suppose that A is an n x n matrix. a. For each eigenvalue λ of A, prove that λ² is an eigenvalue of the matrix A². What is the eigenvector corresponding to X²? b. For each eigenvalue λ of A, prove that X-¹ is an eigenvalue of the matrix A-¹ (you can assume A is invertible). What is the eigenvector corresponding to X-¹?

7. Suppose that A is an n x n matrix. a. For each eigenvalue λ of A, prove that λ² is an eigenvalue of the matrix A². What is the eigenvector corresponding to X²? b. For each eigenvalue λ of A, prove that X-¹ is an eigenvalue of the matrix A-¹ (you can assume A is invertible). What is the eigenvector corresponding to X-¹?

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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7. Suppose that A is an n x n matrix. a. For each eigenvalue λ of A, prove that λ² is an eigenvalue of the matrix A². What is the eigenvector corresponding to X²? b. For each eigenvalue λ of A, prove that X-¹ is an eigenvalue of the matrix A-¹ (you can assume A is invertible). What is the eigenvector corresponding to \¯¹? C. For each eigenvalue λ of A, prove that λ + c is an eigenvalue of the matrix A + CI, where c is a constant number and I is the n × n identity matrix. What is the eigenvector corresponding to λ + c? Prove that the algebraic multiplicity of the corresponding eigenvalue of A is the same as the algebraic multiplicity of the eigenvalue λ + c of A+ cI.
Transcribed Image Text:7. Suppose that A is an n x n matrix. a. For each eigenvalue λ of A, prove that λ² is an eigenvalue of the matrix A². What is the eigenvector corresponding to X²? b. For each eigenvalue λ of A, prove that X-¹ is an eigenvalue of the matrix A-¹ (you can assume A is invertible). What is the eigenvector corresponding to \¯¹? C. For each eigenvalue λ of A, prove that λ + c is an eigenvalue of the matrix A + CI, where c is a constant number and I is the n × n identity matrix. What is the eigenvector corresponding to λ + c? Prove that the algebraic multiplicity of the corresponding eigenvalue of A is the same as the algebraic multiplicity of the eigenvalue λ + c of A+ cI.
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