If invertible matrices A and B have eigenvalues α and β resp. wrt to a common eigenvector x, what is the eigenvalue of
4A3B-4-17AB2 + ∏I
(We'll call this equation (1))
Ax = αx
Bx = βx
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 Ax = αx implies A2x = α2x 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 4α2/β4 - 17αβ2 + ∏
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.