### Problem 8Let \( T \) be a linear operator on a vector space \( V \), and let \( x \) be an eigenvector of \( T \) corresponding to the eigenvalue \( \lambda \). For any positive integer \( m \), prove that \( x \) is an eigenvector of \( T^m \) corresponding to the eigenvalue \( \lambda^m \).

Answered: Image /qna-images/answer/f71a7a8a-45f2-4b5e-9a76-94bbafac0ef8.jpg

Answered: Problem 8 Let T be a linear operator on…

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter4: Eigenvalues And Eigenvectors

Section4.6: Applications And The Perron-frobenius Theorem

Problem 18EQ

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Problem 8 Let T be a linear operator on a vector space V, and let x be an eigenvector of T corresponding to the eigenvalue X. For any positive integer m, prove that x is an eigenvector of Tm corresponding to the eigenvalue Xm