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

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 41EQ

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