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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?

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?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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For an invertible matrix A, prove that A and A have the same eigenyectors., 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 eigenyalue 2. Using matrix operations and the properties of inverse matrices gives which of the following? Ax = Ax A/(Ax) = A/(1x) O(A/A)x = (A/)x Ix = (A/1)x x = A-1x Ax = Ax Ax = 2x AxA-1 = AxA-1 Ax/A = x/A Ax = ix OXAA-1 = A-1x xI = AA-1x x = AA-1x O(A/A)x = ixA-1 Ix = xA-1 A-1Ax = A¬12x Ix = AA-1x x = JA-1x A-1x 1x X = AxA-1 A-lx = 1x A-1x = 1x A-1x = 1x This shows that 1/2 v x is an eigenvector of A with eigenvalue x -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 eigenyectors., 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 eigenyalue 2. Using matrix operations and the properties of inverse matrices gives which of the following? Ax = Ax A/(Ax) = A/(1x) O(A/A)x = (A/)x Ix = (A/1)x x = A-1x Ax = Ax Ax = 2x AxA-1 = AxA-1 Ax/A = x/A Ax = ix OXAA-1 = A-1x xI = AA-1x x = AA-1x O(A/A)x = ixA-1 Ix = xA-1 A-1Ax = A¬12x Ix = AA-1x x = JA-1x A-1x 1x X = AxA-1 A-lx = 1x A-1x = 1x A-1x = 1x This shows that 1/2 v x is an eigenvector of A with eigenvalue x -Select Need Help? Read It 1/x 1/2 Submit Answer
For an invertible matrix A, prove that A and A-l 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 = Àx for a corresponding eigenvalue À. 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/1)x Ix = (A/A)x Ax = 1x Ax = x AXA-1 = ¿xA-1 Ax/A = ix/A Ax = Ax x = JA-1x A-1Ax = A-l1x OXAA-1 = A-1x xI = AA-1x O(A/A)x = xA-1 A-1x = 1x Ix = A-1x Ix = 1xA-1 x = AA-1x x = xA-1 x = A-1x A-1x = 1x A-lx = 1x A-1x = 1x This shows that 1/2 Select- vx is an eigenvector of A-1 with eigenvalue x Need Help? 1/x 1/2
Transcribed Image Text:For an invertible matrix A, prove that A and A-l 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 = Àx for a corresponding eigenvalue À. 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/1)x Ix = (A/A)x Ax = 1x Ax = x AXA-1 = ¿xA-1 Ax/A = ix/A Ax = Ax x = JA-1x A-1Ax = A-l1x OXAA-1 = A-1x xI = AA-1x O(A/A)x = xA-1 A-1x = 1x Ix = A-1x Ix = 1xA-1 x = AA-1x x = xA-1 x = A-1x A-1x = 1x A-lx = 1x A-1x = 1x This shows that 1/2 Select- vx is an eigenvector of A-1 with eigenvalue x Need Help? 1/x 1/2
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