Let A E R have eigenvalues A₁,..., A. Prove that ker(A-AI)= ker((A-AI)2) for each i = 1,...,k if and only if A is diagonalizable. *Hint. The only if direction of this proof is relatively simple, but the if direction is a lot harder. But, the if direction can be made a lot easier if you apply the jordan normal form!

Let A E R have eigenvalues A₁,..., A. Prove that ker(A-AI)= ker((A-AI)2) for each i = 1,...,k if and only if A is diagonalizable. *Hint. The only if direction of this proof is relatively simple, but the if direction is a lot harder. But, the if direction can be made a lot easier if you apply the jordan normal form!

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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