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Determinants and Eigenvalues

by brru25
Tags: determinants, eigenvalues, matrices
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brru25
#1
Oct28-09, 10:09 AM
P: 29
1. The problem statement

For integers m >= n,

Prove det(xIm - AB) = xm-ndet(xIn - 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.
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miqbal
#2
Oct28-09, 02:22 PM
P: 28
Sylvester's determinant theorem covers this: http://en.wikipedia.org/wiki/Determi...minant_theorem

A link to the proof is in the citation.


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