$\ddot{X}+W\dot{X}=X$, $W$ is n-dimensional skew symmetric matrix. $X$ is a column vector and $I$ is identity matrix of appropriate dimension.
\begin{equation} \left(\begin{array} XX \\ \dot{X} \end{array}\right) = e^{\left(\begin{array} 00 & I\\ I& -W \end{array}\right)t} \left(\begin{array} XX(0) \\ \dot{X}(0) \end{array}\right) \end{equation}.
How do I simplify \begin{equation} A = e^{\left(\begin{array} 00 & I\\ I& -W \end{array}\right)t} \end{equation}. For example if euclidean $|W|=1$, Then $e^{Wt}=I+W*Sin(t)+W^2(1-Cos( t))$ (Rogrig's formulae). In the same way how can I simplify the above exponential. I tried doing it by diagonalizing it, but it didnt workout as $W$ is arbitrary. I don't need the complete solution, just point me where to read. I even tried to see if the series converges, but failed. Thankyou in advance
If $W$ is skew-symmetric, they $W=U^*(iD)U$, where $U^*U=I$ and $D$ diagonal, with real elements, and $iD$ has imaginary elements on the diagonal.
Then $$ \ddot X+W\dot X-X=U^*(U\ddot X)+U*iD(U\dot X)-U^*UX=0, $$ or for $Y=UX$ $$ \ddot Y+iD\dot Y-Y=0, $$ Thus $$ Y(t)=\mathrm{e}^{tL_1}Y_1+\mathrm{e}^{tL_2}Y_2, $$ where $$ L_{1,2}=-\frac{i}{2}D+\sqrt{I-\tfrac{1}{4}D^2}. $$ Note that $L_{1,2}$ are readily definable as they are diagonal.