Is the diagonalizing matrix $P$ always diagonalizible?

Question

If $A$ is diagonalizable then $A = PDP^{-1}$ where $D$ is diagonal and $P$ is the diagonalizing matrix whose columns are eigenvectors, is it true that P is always diagonalizible ?

Answer 1

No. Take $A = D = O$, where $O$ is the zero matrix and let $P$ be any nondiagonalisable matrix; this is a counterexample. For instance, if the dimension is two, you can pick $P$ to be $$\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$$