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-? A-x = x Letting x be an eigenvector of A gives Ax = ix for a corresponding eigenvalue 2. Using matrix operations and the properties of inverse matrices gives which of the following? Ax = ix Ax = Ax Ax = ix Ax = ix AXA-1 = ixA-1 OXAA-1 = 2A-x XI = JA-x x = 2A-1x A/(Ax) = A/(ix) Ax/A = ix/A O(A/A)x = (A/2)x Ix = (A/2)x x = JA-1x A-'Ax = A-lax Ix = JA-x x = 1A-x O(A/A)x = ixA-1 Ix = iXA-1 x = ixA-1 = 1x A-1x = 1x A-lx = 1x A-1x = 1x This shows that -Select--- v is an eigenvector of A- with eigenvalue Select---

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-? A-x = x Letting x be an eigenvector of A gives Ax = ix for a corresponding eigenvalue 2. Using matrix operations and the properties of inverse matrices gives which of the following? Ax = ix Ax = Ax Ax = ix Ax = ix AXA-1 = ixA-1 OXAA-1 = 2A-x XI = JA-x x = 2A-1x A/(Ax) = A/(ix) Ax/A = ix/A O(A/A)x = (A/2)x Ix = (A/2)x x = JA-1x A-'Ax = A-lax Ix = JA-x x = 1A-x O(A/A)x = ixA-1 Ix = iXA-1 x = ixA-1 = 1x A-1x = 1x A-lx = 1x A-1x = 1x This shows that -Select--- v is an eigenvector of A- with eigenvalue Select---

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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