The two coupled matrix equations I have are
$$Y_1 = X_1 + AX_1A^T + BX_2B^T$$
$$Y_2 = X_2 + CX_1C^T + DX_2D^T$$
I would like to write them as a single $2 \times 2$ matrix equation where each matrix element consists of the previous matrices.
For example a simpler version of this problem that I can solve is
$$Y_1 = X_1 + AX_1 + BX_2$$
$$Y_2 = X_2 + CX_1 + DX_2$$
which can be rewritten as
$$Y = (I+M)X$$
where
$$Y = \begin{bmatrix} Y_1 & 0 \\ 0 & Y_2 \end{bmatrix}, \qquad M = \begin{bmatrix} A & B \\ C & D \end{bmatrix}, \qquad X = \begin{bmatrix} X_1 & 0 \\ 0 & X_2 \end{bmatrix}$$
and $I$ is the identity matrix.
EDIT:
I don't think the solution to the simple example that I gave is correct. The $X$ and $Y$ in that case should be
$$X = \begin{bmatrix} X_1 \\ X_2 \end{bmatrix}$$
$$Y = \begin{bmatrix} Y_1 \\ Y_2 \end{bmatrix}$$
Best Answer
Your equations can be put together as
$$Y=X+M_0XM_0^T+M_1XM_1^T$$ where the matrices $X$ and $Y$ have been defined in the question and
$$M_0=\begin{bmatrix}A & 0\\0 & D\end{bmatrix}\ \mathrm{and}\ M_1=\begin{bmatrix}0 & B\\C & 0\end{bmatrix}.$$