I am reading a paper, and I am confused about the following equality:
$$\exp\bigg(\begin{bmatrix}0 & P \\ P & 0\end{bmatrix}t\bigg)=\begin{bmatrix} \cosh(Pt) & \sinh(Pt)\\ \sinh(Pt) & \cosh(Pt)\end{bmatrix}$$
where $P\in \mathbb{R}^{n\times n}$.
We know that $$\cosh(P) = \frac{1}{2}[\exp(P)+\exp(-P)],\quad \sinh(P) = \frac{1}{2}[\exp(P)-\exp(-P)]$$
and we know if $P$ is a diagonal matrix, then $\exp(P)$ can be obtained by putting exponential function to each element on the diagonal of $P$.
However, today $P$ is general, how to derive the quality?