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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!

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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!
Transcribed Image Text: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!
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