Matrix exponential when the square is minus the identity
= Matrix exponential when the square is minus the identity
{title2=$e^{tM}=I\cos t+M\sin t$}
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 $\mathbf x'=M\mathbf x$. Consequently every solution has period $2\pi$, and the <eigenvalues> over the complex numbers belong to $\{i,-i\}$.