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For part (a), a 1-eigenvector of A for would also be correct) For part (h

For part (a), a 1-eigenvector of A for would also be correct) For part (h

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.1: Introduction To Eigenvalues And Eigenvectors
Problem 26EQ
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Answer кем For part (a), a 1-eigenvector of A (solution to A = ) is 0 (any nonzero scalar multiple of this vec- A tor would also be correct). For part (b), the 2-eigenspace of A consists of all vectors of the form s 0 [-1/2] [] []} t 0 where s and t are any real numbers, so a basis for this eigenspace is (you 0 could scale each vector to clear fractions if you wanted). Finally, for part (c), A is diagonalizable, and one possible diagonalization of A is -1 -3/2-1/2] [1 0 0] A = PDP-1 = 0 1 0 0 2 0 P-¹. 94 1 0 1 002 Other correct answers are possible; the diagonal entries of D must be 1, 2, and 2 in some order, and the columns of P must be linearly independent eigenvectors of A whose order matches the order of their eigen- values in D. +
Transcribed Image Text:Answer кем For part (a), a 1-eigenvector of A (solution to A = ) is 0 (any nonzero scalar multiple of this vec- A tor would also be correct). For part (b), the 2-eigenspace of A consists of all vectors of the form s 0 [-1/2] [] []} t 0 where s and t are any real numbers, so a basis for this eigenspace is (you 0 could scale each vector to clear fractions if you wanted). Finally, for part (c), A is diagonalizable, and one possible diagonalization of A is -1 -3/2-1/2] [1 0 0] A = PDP-1 = 0 1 0 0 2 0 P-¹. 94 1 0 1 002 Other correct answers are possible; the diagonal entries of D must be 1, 2, and 2 in some order, and the columns of P must be linearly independent eigenvectors of A whose order matches the order of their eigen- values in D. +
Consider the matrix [0-3 A = 0 2 0 2 3 3 You may use, without justification, the fact that the eigenvalues of A are X = 1 and X = 2. (a) Find an eigenvector of A with eigenvalue λ = 1. (b) Find a basis for the eigenspace of A corresponding to the eigenvalue X = 2. (c) Is A diagonalizable? If not, explain why not; if so, find an invertible matrix P and a diagonal matrix D such that A = PDP-1 (you do not have to compute P-¹). -1]
Transcribed Image Text:Consider the matrix [0-3 A = 0 2 0 2 3 3 You may use, without justification, the fact that the eigenvalues of A are X = 1 and X = 2. (a) Find an eigenvector of A with eigenvalue λ = 1. (b) Find a basis for the eigenspace of A corresponding to the eigenvalue X = 2. (c) Is A diagonalizable? If not, explain why not; if so, find an invertible matrix P and a diagonal matrix D such that A = PDP-1 (you do not have to compute P-¹). -1]
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