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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¯±? Letting x be an eigenvector of A gives Ax = ix for a corresponding eigenvalue 1. Using matrix operations and the properties of inverse matrices gives which of the following? Ax = Ax Ax = Ax Ax = Ax Ax = 1x A-lAx = A-12x AXA-1 = ixA-1 Ax/A = Ax/A A/(Ax) = A/(x) OXAA-1 = 2A-lx XI = A-1x x = AA-1x O(A/A)x = ixA-1 Ix = ixA-1 Ix = AA-1x O (A/A)x = (A/2)x Ix = (A/2)x x = AA-1x A-lx = 1x x = AA-1x x = ixA-1 A-1x = 1x A-1x = 1x A-1x = 1x This shows that-Select---v is an eigenvector of A-1 with eigenvalue Select- v Select- Need Help? Read It 1/x 1/2 Submit Answer

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¯±? Letting x be an eigenvector of A gives Ax = ix for a corresponding eigenvalue 1. Using matrix operations and the properties of inverse matrices gives which of the following? Ax = Ax Ax = Ax Ax = Ax Ax = 1x A-lAx = A-12x AXA-1 = ixA-1 Ax/A = Ax/A A/(Ax) = A/(x) OXAA-1 = 2A-lx XI = A-1x x = AA-1x O(A/A)x = ixA-1 Ix = ixA-1 Ix = AA-1x O (A/A)x = (A/2)x Ix = (A/2)x x = AA-1x A-lx = 1x x = AA-1x x = ixA-1 A-1x = 1x A-1x = 1x A-1x = 1x This shows that-Select---v is an eigenvector of A-1 with eigenvalue Select- v Select- Need Help? Read It 1/x 1/2 Submit Answer

Linear Algebra: A Modern Introduction
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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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¯4? 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 = Ax Ax = Ax Ax A-Ax = A-12x AxA-1 = AxA-1 Ax/A = Ax/A = AA-1x XI = A-1x x = AA-1x O(A/A)x = AxA-1 Ix = 1xA-1 x = ixA-1 A/(Ax) = A/(ax) O (A/A)x = (A/2)x Ix = (A/1)x x = 1A-1x Ix = JA-1x OXAA-1 x = AA-1x A-1x = 1x A-1x = 1x A-1x = 1x A-1x = 1x This shows that Select-- v is an eigenvector of A-1 with eigenvalue -Select- v Select Need Help? Read It 1/x 1/2 Submit Answer
Transcribed Image Text: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¯4? 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 = Ax Ax = Ax Ax A-Ax = A-12x AxA-1 = AxA-1 Ax/A = Ax/A = AA-1x XI = A-1x x = AA-1x O(A/A)x = AxA-1 Ix = 1xA-1 x = ixA-1 A/(Ax) = A/(ax) O (A/A)x = (A/2)x Ix = (A/1)x x = 1A-1x Ix = JA-1x OXAA-1 x = AA-1x A-1x = 1x A-1x = 1x A-1x = 1x A-1x = 1x This shows that Select-- v is an eigenvector of A-1 with eigenvalue -Select- v Select Need Help? Read It 1/x 1/2 Submit Answer
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