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 A12 Letting x be an eigenvector of A gives Ax = Ax for a corresponding eigenvalue 2. Using matrix operations and the properties of inverse matrices gives which of the following Ax = Ax Ах Ах XY= Xy Ax = Ax Ax/A = Ax/A AXA-1 AXA-1 A/(Ax) = A/(2x) %3D = AxA-1 Ix = AXA1 x = AxA O(A/A)x = (A/A)x Ix = (A/A)x AAx = A-2x Ix = AA-x x = JA-x OXAA-1 - JA-1x x = AA-x x = A-1x Alx = 1x Ax = 1x A1x = 1x A-lx = 1x This shows that-Select--v is an eigenvector of A with eigenvalue Select-- v

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 A12 Letting x be an eigenvector of A gives Ax = Ax for a corresponding eigenvalue 2. Using matrix operations and the properties of inverse matrices gives which of the following Ax = Ax Ах Ах XY= Xy Ax = Ax Ax/A = Ax/A AXA-1 AXA-1 A/(Ax) = A/(2x) %3D = AxA-1 Ix = AXA1 x = AxA O(A/A)x = (A/A)x Ix = (A/A)x AAx = A-2x Ix = AA-x x = JA-x OXAA-1 - JA-1x x = AA-x x = A-1x Alx = 1x Ax = 1x A1x = 1x A-lx = 1x This shows that-Select--v is an eigenvector of A with eigenvalue Select-- v

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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