Matrix exponential when the square is minus the identity ($e^{tM}=I\cos t+M\sin t$)

If a real square matrix satisfies $M^2=-I$, split the absolutely convergent matrix exponential into even and odd powers to obtain $e^{tM}=I\cos t+M\sin t$. It is the fundamental matrix for ${\mathbf{x}}^{\prime }=M\mathbf{x}$. Consequently every solution has period $2\pi$, and the eigenvalues over the complex numbers belong to $\{i,-i\}$.