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Problem 1. 92x² + (a) If A is diagonalizable, that is A = ODO¹. Assume a polynomial f(x) = ao + ₁x + +ax". Then Prove that f(A) = Q (ƒ(2₂) 0 0 0 0 f(2₂) 0 0 f(2₂)) where 2 eigenvalues of A.

Problem 1. 92x² + (a) If A is diagonalizable, that is A = ODO¹. Assume a polynomial f(x) = ao + ₁x + +ax". Then Prove that f(A) = Q (ƒ(2₂) 0 0 0 0 f(2₂) 0 0 f(2₂)) where 2 eigenvalues of A.

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 1. a₂x² + +anx". Then (a) (b) If A is diagonalizable, that is A = QDQ¹. Assume a polynomial f(x) = a₁ + a₁x + (c) Prove that f(A) = Q (f(2₂) 0 0 f(2₂) 0 0 0 8 ƒ(2₂)) Q¹, where : eigenvalues of A. If f(A) = B, where A and B are both n x n diagonalizable matrices and have identical eigenvectors, then pro that f(ai) = λbi, in which ai and Abi are the eigenvalues of A and B. -7 6 Apply (b) to find the solutions of A that satisfies A² -3A+I = 9. -12 11
Transcribed Image Text:Problem 1. a₂x² + +anx". Then (a) (b) If A is diagonalizable, that is A = QDQ¹. Assume a polynomial f(x) = a₁ + a₁x + (c) Prove that f(A) = Q (f(2₂) 0 0 f(2₂) 0 0 0 8 ƒ(2₂)) Q¹, where : eigenvalues of A. If f(A) = B, where A and B are both n x n diagonalizable matrices and have identical eigenvectors, then pro that f(ai) = λbi, in which ai and Abi are the eigenvalues of A and B. -7 6 Apply (b) to find the solutions of A that satisfies A² -3A+I = 9. -12 11
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