TITLE: Is the product of any two invertible diagonalizable matrices diagonalizable?
QUESTION [16 upvotes]: I'm studying Linear Algebra. I saw an example of a pair of 2 by 2 or n by n diagonalizable matrices, the product of which is not diagonalizable. Is there a similar example when I replace the condition "diagonalizable" by "invertible and diagonalizable"?

REPLY [18 votes]: A counterexample:$$\begin{pmatrix}1&0\\0&-1\end{pmatrix}\begin{pmatrix}1&1\\0&-1\end{pmatrix}=\begin{pmatrix}1&1\\0&1\end{pmatrix}$$
The result is a well known nondiagonalizable matrix, the left matrix of the product is diagonal already, and the right matrix can be written as
$$\begin{pmatrix}1&1\\0&-1\end{pmatrix}=\begin{pmatrix}-1&1\\2&0\end{pmatrix}\begin{pmatrix}-1&0\\0&1\end{pmatrix}\begin{pmatrix}-1&1\\2&0\end{pmatrix}^{-1}.$$