# Eigenvalues of A^-1

1. Jul 15, 2008

### JinM

Is this a correct realization? The eigenspaces corresponding to the eigenvalues of A are the same as the eigenspaces corresponding to the eigenvalues of A^-1, transpose of A, and A^k for any k > 1.

It took me some time to realize this but the v's, when you manipulate these equations, don't change. So I'm lead to believe that the eigenvectors are actually the same for all such variants of A.

2. Jul 15, 2008

### matt grime

Re: Eigenquestion

If A is invertible, then clearly v is an eigenvector for A if and only if it is an eigenvector for A^-1.

Unless you define 'variants' of A then we can't answer your second question. I'll attempt a guess: no, the eigenvectors of A^2 are not the same as eigenvectors of A.

3. Jul 15, 2008

### JinM

Re: Eigenquestion

Sorry for the ambiguity -- you knew what I meant anyway.

Why aren't the eigenvectors of A^k for k > 1 the same as the eigenvectors of A?

4. Jul 15, 2008

### matt grime

Re: Eigenquestion

Why should they be?

5. Jul 15, 2008

### Manchot

Re: Eigenquestion

Assuming that you are referring to an integer k, it's true that eigenvectors of A are automatically eigenvectors of A^k. However, you don't have any guarantee that eigenvectors of A^k are eigenvectors of A.

6. Jul 15, 2008

### JinM

Re: Eigenquestion

Wait a second.

If x is an eigenvalue of A, then x^k is an eigenvalue of A^k (k is an integer).

Are you guys saying that there could possibly be other eigenvalues of A^k that differ from all x^k's (the eigenvalues of A raised to the k)? That's why we can't guarantee matching eigenvectors -- is that it?

Last edited: Jul 16, 2008
7. Jul 16, 2008

### JinM

Re: Eigenquestion

But is that even possible? A^k is always a square matrix, whose order matches that of matrix A.

Hmm..

If we take a 3x3 matrix A with eigenvalues -1, 1, and 4. A^2 will have eigenvalues of 1 (with algebraic multiplicity 2), and 4. A will have three eigenspaces, while A^2 will have two eigenspaces. So, the eigenspace of A corresponding to the eigenvalue -1 will not "live" in the eigenspace of A^1 corresponding to the eigenvalue (-1)^2 = 1. But that contradicts what Manchot is saying.. hrmph

8. Jul 16, 2008

### n_bourbaki

Re: Eigenquestion

No, we're definitely not saying that.

No.

9. Jul 16, 2008

### n_bourbaki

Re: Eigenquestion

What do you mean by order?

You mean 16, not 4, for the e-value of A^2.

Manchot stated that you cannot assume an e-vector of A^k is an e-vector of A for all matrices. You're saying he's wrong just because you can do it for one (diagonalizable) matrix.

Certainly if A is diagonalizable, then e-vectors of A are e-vectors of A^k and vice versa. However, being diagonalizable is a very special property.