I consider the following system of matrix ODEs:

(*)

X'= X*A1 + Y*A2

Y'= -A2^T*X - A1^T*Y,

where X, Y, A1, A2 are matrices in R^[nxn], ^T denotes the transpose.

So far I know that the eigenvalues are symmetric w.r.t. the origin (i.e. s - an eigenvalue => (-s) - an eigenvalue). I'm interested in finding any further properties of the system, especially in connection with the computation of the matrix exponential exp(A*t), where A is obtained through the vectorization of the matrix equations.

The vector form I used is as follows. Let x and y be the vectors obtained by stacking the columns of X and Y. Let z=(x^T y^T)^T. Then (*) can be rewritten as

z'=Lz,

where

L=[kron(A1',I) kron(A2',I); -kron(I,A2') -kron(I,A1')],

written in Matlab notation with kron(A,B) denoting the Kronecker product of A and B.

All references/suggestions will be of great help.

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