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this problem, we consider the diagonalization of a symmetric matrix. a. Prove that the n x n matrix A is orthogonal if and only if the columns (rows) of A form an orthonormal set. 11

this problem, we consider the diagonalization of a symmetric matrix. a. Prove that the n x n matrix A is orthogonal if and only if the columns (rows) of A form an orthonormal set. 11

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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In this problem, we consider the diagonalization of a symmetric matrix. a. Prove that the \( n \times n \) matrix \( A \) is orthogonal if and only if the columns (rows) of \( A \) form an orthonormal set. b. Prove that if \( A \) is a symmetric \( n \times n \) matrix, then there exists an orthogonal matrix \( P \) such that \( P^{-1}AP = P^TAP = D \), where \( D \) is a diagonal matrix. Furthermore, the eigenvalues of \( A \) lie on the diagonal of \( D \).
Transcribed Image Text:In this problem, we consider the diagonalization of a symmetric matrix. a. Prove that the \( n \times n \) matrix \( A \) is orthogonal if and only if the columns (rows) of \( A \) form an orthonormal set. b. Prove that if \( A \) is a symmetric \( n \times n \) matrix, then there exists an orthogonal matrix \( P \) such that \( P^{-1}AP = P^TAP = D \), where \( D \) is a diagonal matrix. Furthermore, the eigenvalues of \( A \) lie on the diagonal of \( D \).
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