- #1
jon555
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Hi I have this question for my Linear Algebra class and I can't seem to figure it out.
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 can't seem to get it. I think I am close but I can't seem to make the connection.
Thank you in advance for any help you can give
Jon
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.
----------------------------------------
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 can't seem to get it. I think I am close but I can't seem to make the connection.
Thank you in advance for any help you can give
Jon