|Oct28-09, 10:09 AM||#1|
Determinants and Eigenvalues
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.
|Oct28-09, 02:22 PM||#2|
Sylvester's determinant theorem covers this: http://en.wikipedia.org/wiki/Determi...minant_theorem
A link to the proof is in the citation.
|determinants, eigenvalues, matrices|
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