I was wondering whether the classical procedure of solving a system of linear homogeneous differential equations by finding the eigenvalues and eigenvectors is applicable to Matrix-equations as well, meaning I am having the following problem:
Find a $2n\times n$ matrix $X$ such that
$ X'=AX$
where $A=\begin{bmatrix}0&-E \\ K & 0 \end{bmatrix}$, $E=diag(1,..,1)$ denotes the identity matrix in dimension $n$ and $K$ is a symmetric positive semi-definite $n\times n$-matrix.
Since this is my first question here, please don't be harsh on any typos etc. Many thanks in advance!
As a first step it is nice to reduce the problem to a problem with square matrices. Thus you can try to split $X$ into two $n\times n$ matrices and rewrite your problem. Then you can define the exponential of a matrix for square matrices which is the general solution for such problems.