In general when one proves that the product of semisimple (i.e. diagonalizable) matrices is semisimple one assumes they commute and are thus simultaneously diagonalizable, and then the result follows. I was wondering if anyone knew of an example of noncommuting semisimple matrices whose product is not semisimple.

How about $\begin{pmatrix} 5 & 3 \\\ 8 & 5 \end{pmatrix}$ and $\begin{pmatrix} 2 & 3 \\\ 3 & 5 \end{pmatrix}$? 


$\begin{pmatrix}1&0\\0&0\end{pmatrix}\begin{pmatrix}0&1\\1&0\end{pmatrix}=\begin{pmatrix}0 & 1\\0&0\end{pmatrix}$ 

