Stacking Matrix Equations
Suppose $\mathbf x_1,\mathbf x_2,\mathbf y_1,\mathbf y_2$ are $1 \times n$ vectors and $\mathbf A_1,\mathbf A_2$ are $n \times n$ matrices and consider the system \begin{align} \mathbf x_1 =&\;\; \omega_1\mathbf x_2 \mathbf A_1 + (1-\omega_1)\mathbf y_1\\[2ex] \mathbf x_2 =&\;\; \omega_2\mathbf x_1 \mathbf A_2 + (1-\omega_2)\mathbf y_2. \end{align} where $\omega_1,\omega_2$ are scalars. Is it possible to stack these equations to solve for $$ \mathbf X = \begin{bmatrix} \mathbf x_1 \\ \mathbf x_2 \end{bmatrix}?$$
$\endgroup$ 21 Answer
$\begingroup$Is this what you want? $$\begin{bmatrix} I_n &-\omega_1 A_1 \\ -\omega_2 A_2 & I_n \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} (1-\omega_1)y_1 \\ (1-\omega_2)y_2\end{bmatrix} = \begin{bmatrix} (1-\omega_1)I_n & 0 \\ 0 & (1-\omega_2)I_n \end{bmatrix}\begin{bmatrix}y_1\\ y_2\end{bmatrix}$$
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