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4. True or False? If true, prove. If false, provide a counterexample. a. If same size matrices A and B have exactly the same eigenvalues (including multiplicities), they must be similar. b. If a square matrix A is diagonalizable, so is Ak, for any positive integer k. C. If a square invertible matrix A is diagonalizable, so is A-¹. d. If a square matrix A is invertible, then it's diagonalizable. e. For any m x n matrix A, the matrix AT A is orthogonally diagonalizable. f. Let A be a symmetric n x n matrix. Suppose ₁ and ₂ are eigenvectors corresponding to two distinct eigenvalues of A. Then, ||x₁ + x₂ || = ||X₁|| + ||x₂||. g. If an n x n matrix A has only real eigenvalues, then it must be symmetric.

4. True or False? If true, prove. If false, provide a counterexample. a. If same size matrices A and B have exactly the same eigenvalues (including multiplicities), they must be similar. b. If a square matrix A is diagonalizable, so is Ak, for any positive integer k. C. If a square invertible matrix A is diagonalizable, so is A-¹. d. If a square matrix A is invertible, then it's diagonalizable. e. For any m x n matrix A, the matrix AT A is orthogonally diagonalizable. f. Let A be a symmetric n x n matrix. Suppose ₁ and ₂ are eigenvectors corresponding to two distinct eigenvalues of A. Then, ||x₁ + x₂ || = ||X₁|| + ||x₂||. g. If an n x n matrix A has only real eigenvalues, then it must be symmetric.

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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**4. True or False? If true, prove. If false, provide a counterexample.** a. If same size matrices \(A\) and \(B\) have exactly the same eigenvalues (including multiplicities), they must be similar. b. If a square matrix \(A\) is diagonalizable, so is \(A^k\), for any positive integer \(k\). c. If a square invertible matrix \(A\) is diagonalizable, so is \(A^{-1}\). d. If a square matrix \(A\) is invertible, then it’s diagonalizable. e. For any \(m \times n\) matrix \(A\), the matrix \(A^T A\) is orthogonally diagonalizable. f. Let \(A\) be a symmetric \(n \times n\) matrix. Suppose \(\mathbf{x}_1\) and \(\mathbf{x}_2\) are eigenvectors corresponding to two distinct eigenvalues of \(A\). Then, \(\|\mathbf{x}_1 + \mathbf{x}_2\| = \|\mathbf{x}_1\| + \|\mathbf{x}_2\|\). g. If an \(n \times n\) matrix \(A\) has only real eigenvalues, then it must be symmetric.
Transcribed Image Text:**4. True or False? If true, prove. If false, provide a counterexample.** a. If same size matrices \(A\) and \(B\) have exactly the same eigenvalues (including multiplicities), they must be similar. b. If a square matrix \(A\) is diagonalizable, so is \(A^k\), for any positive integer \(k\). c. If a square invertible matrix \(A\) is diagonalizable, so is \(A^{-1}\). d. If a square matrix \(A\) is invertible, then it’s diagonalizable. e. For any \(m \times n\) matrix \(A\), the matrix \(A^T A\) is orthogonally diagonalizable. f. Let \(A\) be a symmetric \(n \times n\) matrix. Suppose \(\mathbf{x}_1\) and \(\mathbf{x}_2\) are eigenvectors corresponding to two distinct eigenvalues of \(A\). Then, \(\|\mathbf{x}_1 + \mathbf{x}_2\| = \|\mathbf{x}_1\| + \|\mathbf{x}_2\|\). g. If an \(n \times n\) matrix \(A\) has only real eigenvalues, then it must be symmetric.
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