Let A be an invertible matrix. Show that if v is an eigenvector of A belonging to the eigenvalue A, then v is also an eigenvector of A-1 belonging to the eigenvalue 1/A. (In particular, 1/A is an eigenvalue of A-1.)

Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter7: Distance And Approximation
Section7.4: The Singular Value Decomposition
Problem 26EQ
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hint: : Use the definitions of eigenvalue and eigenvector, and not the characteristic polynomial.

Let A be an invertible matrix. Show that if v is an eigenvector of A belonging to the
eigenvalue A, then v is also an eigenvector of A- belonging to the eigenvalue 1/A. (In
particular, 1/A is an eigenvalue of A-1.)
Transcribed Image Text:Let A be an invertible matrix. Show that if v is an eigenvector of A belonging to the eigenvalue A, then v is also an eigenvector of A- belonging to the eigenvalue 1/A. (In particular, 1/A is an eigenvalue of A-1.)
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