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 non-commuting semisimple matrices whose product is not semisimple. 



The product of non-commuting semisimple matrices need not be semisimple
How about $\begin{pmatrix} 5 & 3 \\\ 8 & 5 \end{pmatrix}$ and
 $\begin{pmatrix} 2 & -3 \\\ -3 & 5 \end{pmatrix}$?


