For an invertible matrix A, prove that A and A-1 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 = Ax for a corresponding eigenvalue A. Using matrix operations and the properties of inverse matrices gives which of the following? Ax = ix Ax = ix Ax = ix Ax = ix A-Ax = A-1x A/(Ax) A/(Ax) AxA -1 Ax/A = Ax/A %3D O(A/A)x = (A/2)x Ix = (A/A)x x = LA-x 1x OXAA-1 = LA-x xl = 1A-x x = LA-x O(A/A)x Ix = ixA1 %3! Ix = = lA-x x = A-1x A-x = 1x x = ixA-1 A-1x = A-x = 1x This shows that -Select--- V is an eigenvector of A with eigenvalue --Select--

For an invertible matrix A, prove that A and A-1 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 = Ax for a corresponding eigenvalue A. Using matrix operations and the properties of inverse matrices gives which of the following? Ax = ix Ax = ix Ax = ix Ax = ix A-Ax = A-1x A/(Ax) A/(Ax) AxA -1 Ax/A = Ax/A %3D O(A/A)x = (A/2)x Ix = (A/A)x x = LA-x 1x OXAA-1 = LA-x xl = 1A-x x = LA-x O(A/A)x Ix = ixA1 %3! Ix = = lA-x x = A-1x A-x = 1x x = ixA-1 A-1x = A-x = 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 24EQ

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