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

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.
