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 2. Using matrix operations and the properties of inverse matrices gives which of the following? Ax = Ax A/(Ax) = A/(Ax) O(A/A)x = (A/)x Ax = Ax Ax = Ax Ax = 1x = AXA-1 = 1A-1x xI = JA-1x x = AA-x A-'Ax = A-1ax Ax/A = Ax/A AXA-1 O(A/A)x = ixA-1 Ix = ixA-1 Ix = JA-1x Ix = (A/A)x OXA4-1 x = 1A-1x x = 1A-1x A-1x = 1x x = 1XA-1 A-1x = 1x A-1x = 1x A-1x = 1x This shows that --Select--- v is an eigenvector of A1 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 A1? 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 A/(Ax) = A/(Ax) O(A/A)x = (A/)x Ax = Ax Ax = Ax Ax = 1x = AXA-1 = 1A-1x xI = JA-1x x = AA-x A-'Ax = A-1ax Ax/A = Ax/A AXA-1 O(A/A)x = ixA-1 Ix = ixA-1 Ix = JA-1x Ix = (A/A)x OXA4-1 x = 1A-1x x = 1A-1x A-1x = 1x x = 1XA-1 A-1x = 1x A-1x = 1x A-1x = 1x This shows that --Select--- v is an eigenvector of A1 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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