Proving Eigenspace Correspondence for Similar Matrices

[jon555]

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

 

[CompuChip]

Probably this is the type of question you're going to slap yourself in the head for, when seeing how straightforward it is.

So you have B = P^-1AP. I'll be using c instead of lambda because I'm too lazy to find a HTML lambda. You want to prove that
Av = cv, if and only if B(P^-1 v) = c (P^-1 v).

So start by the direct implication: suppose that Av = cv. Now calculate B(P^-1 v) using what you know about B.

 

[HallsofIvy]

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).

Ahhh! "let lambda be an eigenvalue". I read this as "let lambda not equal an eigenvalue" and couldn't understand what you were saying from here on!

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