Matrix $A$ is given in the form,
$ A= \begin{bmatrix} 0 & X \\ Y & 0 \end{bmatrix} $
where $X$ and $Y$ are symmetric $n\times n$ matrices. Why A is diagonalizable?
Note: I think this form of A always appear when writing a second order ODE system into two coupled first order ODEs. It is important when dealing with systems with complex spectrum.
I think the matrix $A = \begin{bmatrix} 0 & X \\ Y & 0 \end{bmatrix}$ is similar to $B= \frac{1}{2}\begin{bmatrix} X+Y & 0 \\ 0 & -X-Y \end{bmatrix}$, or something nearby to that. If the matrices $X$ and $Y$ are symmetric, then so is the matrix $B$. Therefore, $B$ is diagonalizable and it follows that the similar matrix $A$ is diagonalizeable too.