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 A1? 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 = Ax Ax = Ax Ax = Ax Ax = Ax A / (Ax) = A / (Ax) O(A / A)x = (A / A)x Ix = (A / A)x x = AA-1x 1x AxA-1 = AxA-1 Ax / A = Ax / A A-Ax = A-1Ax OXAA-1 = NA-1x xl = NA-1x x = NA-x O(A / A)x = AxA-1 Ix = AxA-1 Ix = AA-1x x = AA-1x x = AxA-1 A-x = x A-x = 1x A-1x = A-x = 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 A1? 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 = Ax Ax = Ax Ax = Ax Ax = Ax A / (Ax) = A / (Ax) O(A / A)x = (A / A)x Ix = (A / A)x x = AA-1x 1x AxA-1 = AxA-1 Ax / A = Ax / A A-Ax = A-1Ax OXAA-1 = NA-1x xl = NA-1x x = NA-x O(A / A)x = AxA-1 Ix = AxA-1 Ix = AA-1x x = AA-1x x = AxA-1 A-x = x A-x = 1x A-1x = A-x = 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 24EQ

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