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? Letting x be an eigenvector of A gives Ax = 1x for a corresponding eigenvalue 2. Using matrix operations and the properties of inverse matrices gives which of the following? Ax = Ax Ax = Ax Ax = Ax AxA-1 = AxA-1 = A-1x Ax = ix A-1Ax = A-1x Ix = AA-1x Ax/A = Ax/A A/(Ax) = A/(Ax) (A/A)x = (A/2)x Ix = (A/A)x x = AA-1x OXAA-1 xI = 1A-1x x = AA-1x O(A/A)x = ixA-1 Ix = ixA-1 x = AA-1x x = AxA-1 1x A-1x = 1x A-1x = 1x A-1x = 1x A-1x = This shows that ---Select--- vis an eigenvector of A-1 with eigenvalue --Select--- v Select-- Need Help? 1/x 1/2 ed Work Revert to Last Response
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? Letting x be an eigenvector of A gives Ax = 1x for a corresponding eigenvalue 2. Using matrix operations and the properties of inverse matrices gives which of the following? Ax = Ax Ax = Ax Ax = Ax AxA-1 = AxA-1 = A-1x Ax = ix A-1Ax = A-1x Ix = AA-1x Ax/A = Ax/A A/(Ax) = A/(Ax) (A/A)x = (A/2)x Ix = (A/A)x x = AA-1x OXAA-1 xI = 1A-1x x = AA-1x O(A/A)x = ixA-1 Ix = ixA-1 x = AA-1x x = AxA-1 1x A-1x = 1x A-1x = 1x A-1x = 1x A-1x = This shows that ---Select--- vis an eigenvector of A-1 with eigenvalue --Select--- v Select-- Need Help? 1/x 1/2 ed Work Revert to Last Response
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Transcribed Image Text: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 A1?
Letting x be an eigenvector of A gives Ax = Ax 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 = Ax
AxA-1 = 1xA-1
Ax/A = Ax/A
A/(Ax) = A/(1x)
(A/A)x = (A/1)x
A-lAx = A-12x
Ix = AA-1x
x = 1A-1x
= AA-1x
(A/A)x
= 1xA-1
OXAA-1
Ix = (A/1)x
xI = AA-1x
Ix = AxA-1
x = 1A-1x
1x
x = 1A-1x
x = 1xA-1
A-1x =
1x
1x
A-1x = 1x
A-1x =
A-1x
This shows that
-Select--- v is an eigenvector of A-1 with eigenvalue
Select--- v
Select-
Need Help? 1/x
1/1
d Work Revert to Last Response
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
