It is asked to calculate $e^{tA}$, where
$$A=\begin{pmatrix} 0&1 & 0&0 \\ 3\omega ^2&0 &0 &2 \omega \\ 0& 0 & 0 &1 \\ 0& -2 \omega &0 &0 \end{pmatrix}$$
First, I've tried to calculate $A^2$ and $A^3$ to see if it has a "known form" or if the matrix is nilpotent. It doesn't seen the case. Then, I've tried Jordan Decomposition of A , which looks like, for me, the best way of doing it. It has 2 imaginary eigenvalues $-i\omega, \; i\omega$ and two real $0$ with multiplicity 2. It gives me the answer (I mean, is not incorrect), but I am wondering if there is a better way of calculating this.
Thanks in advance!
I think computation via some kind of Jordan decomposition is pretty much the best you could do for calculation by hand.
I suspect that you might make things slightly easier if you used something like Jordan real form, if you're comfortable exponentiating rotation matrices.