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## Homework Statement

If invertible matrices A and B have eigenvalues α and β resp. wrt to a common eigenvector

**x**, what is the eigenvalue of

4A

^{3}B

^{-4}-17AB

^{2}+ ∏I

(We'll call this equation (1))

## Homework Equations

A

**x**= α

**x**

B

**x**= β

**x**

## The Attempt at a Solution

I think I'm ok for the actual eigenvalue. Basically we "exchange" α for A and β for B in the equation above, since A

**x**= α

**x**implies A

^{2}

**x**= α

^{2}

**x**and so on and so forth. ∏I is just an identity matrix multiplied by ∏, so it's eigenvalue is ∏ (with multiplicity two).

So I get 4α

^{2}/β

^{4}- 17αβ

^{2}+ ∏

But I can't seem to figure out how to approach getting the corresponding Eigenvector. I know the characteristic equation

det(A - λI) = 0

may tell me something, but if I plug in (1) for A and the eigenvalue above for λ I just get a complete mess.

-Dave K