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eigenvalue

I need help with this problem. I've been working on it for a while and unable to solve it. I need to understand how to solve this problem through examples. With step by step breakdown to fully complete the problem.

Thank you for your help it is greatly appreciated!

(a) Prove that if A and B are both invertible n x n matrices, then AB and BA have the same eigenvalues.

(b) Given that A is an n x n matrix with eigenvalues λi, i = 1, . . . , n, prove that:
n n
(i) ∑λi = tr (A) and (ii) ∏ λi = │A│
i=1 i=1

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(a) Prove that if A and B are both invertible n x n matrices, then AB and BA have the same eigenvalues.

One definition of an eigenvalue of an nxn matrix M, is that it is the root of this "characteristic" polynomial PM(λ):

PM (λ)= det(λIn-M).

Obviously, if AB and BA have the same characteristic polynomial, then the roots are the same, so the eigenvalues are the ...

Solution Summary

The expert examines an eigenvalue matrix.

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