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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 \ 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. (a) A = 3 22 for F = R 0 -2 -3 (b) A = -1 1 -1 for F= R 2 2 5 (c) A = (2 ½) for F = C 20 -1 (d) A = 41 -4 for F = R 2 -1 Chap. 5 Diagonalization

258 4. For each of the following matrices A € Mnxn (F), (i) Determine all the eigenvalues of A. (ii) For each eigenvalue \ 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. (a) A = 3 22 for F = R 0 -2 -3 (b) A = -1 1 -1 for F= R 2 2 5 (c) A = (2 ½) for F = C 20 -1 (d) A = 41 -4 for F = R 2 -1 Chap. 5 Diagonalization

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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Section 5.1:4(b, d) only!

 

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 Qand a diagonal matrix D such that Q¹AQ = D. 258 1 (a) A = ( 3 ) for FR 0 -2 3 (b) A = -1 1 -1 for FR 2 2 5 i 1 (c) A= = for F = C = 2 0 - (d) A= 4 1 -4 for F = R 2 0 -1 Chap. 5 Diagonalization
Transcribed Image Text: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 Qand a diagonal matrix D such that Q¹AQ = D. 258 1 (a) A = ( 3 ) for FR 0 -2 3 (b) A = -1 1 -1 for FR 2 2 5 i 1 (c) A= = for F = C = 2 0 - (d) A= 4 1 -4 for F = R 2 0 -1 Chap. 5 Diagonalization
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