Eigen-values of $AB$ and $BA$?

Question

Let $A,B \in M(n,\mathbb{C})$ be two $n\times n$ matrices. I would like know how to prove that eigen-value of $AB$ is the same as the eigen-values of $BA$.

A related question. Not an exact duplicate, because there it was assumed that $B$ is invertible. Studying the answers given there will get you started anyway (more or less along the lines of Matt Pressland's +1 answer). — Jyrki Lahtonen, Dec 19 '12 at 14:10

score 6 · Answer 1 · edited Dec 19 '12 at 14:11

6

Hint: let $v$ be an eigenvector of $AB$ with eigenvalue $\lambda$. What is $BABv$?

edited Dec 19 '12 at 14:11

Jyrki Lahtonen

133,153

answered Dec 19 '12 at 11:48

mdp

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I think you mean $Bv$ – Tomas Dec 19 '12 at 13:52
@sun Nope, although that would also be a good question to ask. Answering the question I asked should lead to an answer to your suggested modification, if I've understood correctly. – mdp Dec 19 '12 at 13:59
right. I initially thought you wanna ask what's the eigenvector of BA. – Tomas Dec 19 '12 at 14:15

score 3 · Answer 2 · answered Dec 19 '12 at 14:07

3

you can prove $|\lambda I-AB|=|\lambda I-BA|$ by computing the determinant of following $$ \left( \begin{array}{cc} I & A \\ B & I \\ \end{array} \right) $$ in two diffeerent ways.

answered Dec 19 '12 at 14:07

Tomas

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Eigen-values of $AB$ and $BA$?

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