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For an invertible matrix A, prove thatA and Ahave the same eigenvectors. How are the eigenvalues of A related to the eigenvalues ofA5 Letting x be an eigenvector of A gives Ax = Ax for a corresponding eigenvalue A. Using matnx operations and the properties of inverse matrices gives which of the following? 華 華 Ax = Ax Ax = Ax Ax = Ax Ax = Ax A/(Ax) = A/(Ax) Ax/A = ix/A AxA = AXA-1 AAx = Ax Ix = Mx x = AAx o(4/A)x = (A/A)x Ix = (A/A)x Ix = AxA-1 x = Ax x = AAx x AAx Ax = 1x A-x = 1x This shows that Select- v35 an eigenvector of A with eigenvalue-Select- Select--- Need Help? 1/x

For an invertible matrix A, prove thatA and Ahave the same eigenvectors. How are the eigenvalues of A related to the eigenvalues ofA5 Letting x be an eigenvector of A gives Ax = Ax for a corresponding eigenvalue A. Using matnx operations and the properties of inverse matrices gives which of the following? 華 華 Ax = Ax Ax = Ax Ax = Ax Ax = Ax A/(Ax) = A/(Ax) Ax/A = ix/A AxA = AXA-1 AAx = Ax Ix = Mx x = AAx o(4/A)x = (A/A)x Ix = (A/A)x Ix = AxA-1 x = Ax x = AAx x AAx Ax = 1x A-x = 1x This shows that Select- v35 an eigenvector of A with eigenvalue-Select- Select--- Need Help? 1/x

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 thatA and 4 have the same eigenvectars. How are the eigenvalues of A related to the eigenvalues ofA Letting x be an eigenvector of A gives Ax = Ax for a corresponding eigenvalue. Using matrix operations and the properties of inverse matrices gives which of the following? Ax = Ax Ax/A = Ax/A Ax = Ax Ax = Ax Ax = Ax AAx = AAx A/(Ax) = A/(Ux) AxA = AxA Ix = AAx x = AAx OA/A)x = (A/A)x Ix = (A/A)x x = AA¯-x xI = JAx x = Ax4 A = 1x A-x = 1x Ax = 1x 4-x - 1x This shows that Select v3s an eigenvector of A* with eigenvalue -Select- Select- Need Help? 1/x 1/1
Transcribed Image Text:For an invertible matrix A, prove thatA and 4 have the same eigenvectars. How are the eigenvalues of A related to the eigenvalues ofA Letting x be an eigenvector of A gives Ax = Ax for a corresponding eigenvalue. Using matrix operations and the properties of inverse matrices gives which of the following? Ax = Ax Ax/A = Ax/A Ax = Ax Ax = Ax Ax = Ax AAx = AAx A/(Ax) = A/(Ux) AxA = AxA Ix = AAx x = AAx OA/A)x = (A/A)x Ix = (A/A)x x = AA¯-x xI = JAx x = Ax4 A = 1x A-x = 1x Ax = 1x 4-x - 1x This shows that Select v3s an eigenvector of A* with eigenvalue -Select- Select- Need Help? 1/x 1/1
For an invertible matrix A, prove that A and A have the same eigenvectors. How are the eigenvalues of A related to the eigenvalues ofA, Letting x be an eigenvector of A gives Ax = Ax for a coesponding eigenvalue . Using matrix operations and the properties of inverse matrices gives which of the following? Ax Ax Ax = Ax Ax = Ax Ax = Ax 4x/A= Ax/A AxA = ÀxA 1 A Ax = AÀx 7x = Mx A/(Ax) = A/(Ax) oA/A)x = (A/a)x Zx = (A/Ajx x = AAx (4/A)x = hKA! XAA x/ = AAx A = 1x 4Tx = -x A¯x = 1x This shows that Select-- is an eigenvector of Awith eigenvalue-Select v -Select--
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 ofA, Letting x be an eigenvector of A gives Ax = Ax for a coesponding eigenvalue . Using matrix operations and the properties of inverse matrices gives which of the following? Ax Ax Ax = Ax Ax = Ax Ax = Ax 4x/A= Ax/A AxA = ÀxA 1 A Ax = AÀx 7x = Mx A/(Ax) = A/(Ax) oA/A)x = (A/a)x Zx = (A/Ajx x = AAx (4/A)x = hKA! XAA x/ = AAx A = 1x 4Tx = -x A¯x = 1x This shows that Select-- is an eigenvector of Awith eigenvalue-Select v -Select--
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