Hi I have this question for my Linear Algebra class and I can't seem to figure it out.(adsbygoogle = window.adsbygoogle || []).push({});

Let A and B be n x n matrices such that B = (P^-1)AP and let lambda ne an eigenvalue of A (and hence of B). Prove the following results:

(a) A vector b in R^n is in the eigenspace of A corresponding to labmda if and only if (P^-1)v is in the eigenspace corresponding to lambda.

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My thought process was that since A and B are similar they will will have the same characteristic polynomial, eigenvalues, multiplicity etc. And B is the diagonal matrix of A and the columns of P are a basis for R^n. Also the equation Ax=b is consistent for every b in R^n if A is invertible.

Ive been working on this problem all week and cant seem to get it. I think Im close but I cant seem to make the connection.

Thank you in advance for any help you can give

Jon

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# Diagonal Matrix Proof help

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