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258 4. For each of the following matrices A € Mnxn (F), (i) Determine all the eigenvalues of A. (ii) For each eigenvalue A of A, find the set of eigenvectors correspond- ing to A. (iii) If possible, find a basis for F" consisting of eigenvectors of A. (iv) If successful in finding such a basis, determine an invertible matrix Q and a diagonal matrix D such that Q¹AQ = D. Chap. 5 Diagonalization 1 2 (a) A = for F = R 3 2 0-2 (b) A = = -1 1 -1 for F = R 2 2 5 (c) A = ( 2 ) for F = C (d) A = 20 41 20-1 -4 for F = R

258 4. For each of the following matrices A € Mnxn (F), (i) Determine all the eigenvalues of A. (ii) For each eigenvalue A of A, find the set of eigenvectors correspond- ing to A. (iii) If possible, find a basis for F" consisting of eigenvectors of A. (iv) If successful in finding such a basis, determine an invertible matrix Q and a diagonal matrix D such that Q¹AQ = D. Chap. 5 Diagonalization 1 2 (a) A = for F = R 3 2 0-2 (b) A = = -1 1 -1 for F = R 2 2 5 (c) A = ( 2 ) for F = C (d) A = 20 41 20-1 -4 for F = R

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.3: Eigenvalues And Eigenvectors Of N X N Matrices
Problem 24EQ
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258 4. For each of the following matrices A Є Mnxn (F), (i) Determine all the eigenvalues of A. (ii) For each eigenvalue A of A, find the set of eigenvectors correspond- ing to A. (iii) If possible, find a basis for F" consisting of eigenvectors of A. (iv) If successful in finding such a basis, determine an invertible matrix Q and a diagonal matrix D such that Q¹AQ = D. Chap. 5 Diagonalization 1 2 (a) A= for F = R 3 2 0 -2 (b) A = = 1 -1 for F = R 2 2 5 (c) A= (2 ½) for F = C 2 0 -1 (d) A 4 1 -4 for FR 0 -1
Transcribed Image Text:258 4. For each of the following matrices A Є Mnxn (F), (i) Determine all the eigenvalues of A. (ii) For each eigenvalue A of A, find the set of eigenvectors correspond- ing to A. (iii) If possible, find a basis for F" consisting of eigenvectors of A. (iv) If successful in finding such a basis, determine an invertible matrix Q and a diagonal matrix D such that Q¹AQ = D. Chap. 5 Diagonalization 1 2 (a) A= for F = R 3 2 0 -2 (b) A = = 1 -1 for F = R 2 2 5 (c) A= (2 ½) for F = C 2 0 -1 (d) A 4 1 -4 for FR 0 -1
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