Determinants and Eigenvalues

brru25

1. The problem statement

For integers m >= n,

Prove det(xI_m - AB) = x^m-ndet(xI_n - BA) for any x in R.

2. Relevant equations

A is an m x n matrix
B is an n x m matrix

3. The attempt at a solution

I tried working out the characteristic polynomials by hand but it just seems too tedious for a nice proof. I know that each x is an eigenvalue of AB but after that I'm stumped.

miqbal

Sylvester's determinant theorem covers this: http://en.wikipedia.org/wiki/Determi...minant_theorem

A link to the proof is in the citation.