Rewriting two coupled linear matrix equations as a single linear matrix equation

Question

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 $$X = \begin{bmatrix} X_1 \\ X_2 \end{bmatrix}$$ $$Y = \begin{bmatrix} Y_1 \\ Y_2 \end{bmatrix}$$

Answer 1

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

$$M_0=\begin{bmatrix}A & 0\\0 & D\end{bmatrix}\ \mathrm{and}\ M_1=\begin{bmatrix}0 & B\\C & 0\end{bmatrix}.$$

Answer 2

$\def\v{\operatorname{vec}}\def\m#1{\left[\begin{array}{c}#1\end{array}\right]}$You could define the vectors $$\eqalign{ x_k &= \v(X_k) \quad &y_k = \v(Y_k) \\ x &= \m{x_1\\x_2} \quad &y = \m{y_1\\y_2} \\ }$$ and vectorize the equations using Kronecker products to obtain $$\eqalign{ {\cal K} &= \m{(I+A\otimes A)&B\otimes B\\C\otimes C&(I+D\otimes D)}\\ y &= {\cal K} x \\ }$$