This is a question from Serre's Exercise book in Matrix theory. I don't even know how to start. Any help would be appreciated.
Assume that the characteristic of the field $k$ is not equal to 2. Given $M \in GL_n(k)$, show that the matrix \begin{align} \begin{pmatrix}0_n & M^{-1} \\ M & 0_n\end{pmatrix} \end{align}
is diagonalizable. Find its eigenvectors and eigenvalues. More generally, show that every involution ($A^2=I$) is diagonalizable.