For an invertible matrix A, prove that A and A- have the same eigenvectors. How are the eigenvalues of A related to the eigenvalues of A-1? Letting x be an eigenvector of A gives Ax = ix for a corresponding eigenvalue 1. Using matrix operations and the properties of inverse matrices gives which of the following? Ax = ix Ах - Ах A/(Ax) = A/(1x) O(A/A)x = (A/a)x Ix = (A/a)x x = 1A-x Ax = ix Ax = ix Ax/A = ix/A (A/A)x = ixA-1 Ix = ixA-1 x = ixA-1 A'Ax = A-1ax Ix = 1A-1x AxA-1 = ixA-1 = A-'x OXA4-1 XI = A-1x x = 1A-1x A-x = 1x x = 1A-x A-x = 1x A-1x = 1x A-ix = 1x This shows that -Select-- v is an eigenvector of A¬1 with eigenvalue -Select---v -Select- -Select-- 1/x 1/x 1/2 1/2

For an invertible matrix A, prove that A and A- have the same eigenvectors. How are the eigenvalues of A related to the eigenvalues of A-1? Letting x be an eigenvector of A gives Ax = ix for a corresponding eigenvalue 1. Using matrix operations and the properties of inverse matrices gives which of the following? Ax = ix Ах - Ах A/(Ax) = A/(1x) O(A/A)x = (A/a)x Ix = (A/a)x x = 1A-x Ax = ix Ax = ix Ax/A = ix/A (A/A)x = ixA-1 Ix = ixA-1 x = ixA-1 A'Ax = A-1ax Ix = 1A-1x AxA-1 = ixA-1 = A-'x OXA4-1 XI = A-1x x = 1A-1x A-x = 1x x = 1A-x A-x = 1x A-1x = 1x A-ix = 1x This shows that -Select-- v is an eigenvector of A¬1 with eigenvalue -Select---v -Select- -Select-- 1/x 1/x 1/2 1/2

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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