Suppose the matrices $A$ and $B$ are $M \times N$; $X_1$ and $X_2$ are $N \times 1$; $G_1$ and $G_2$ are $M \times 1$
Then I hope to convert the equation
$$ \begin{equation} \begin{bmatrix} A&B\\B&A \end{bmatrix} \begin{bmatrix} X_1\\X_2 \end{bmatrix} = \begin{bmatrix} G_1\\G_2 \end{bmatrix} \end{equation} $$ to the form for applying the specific algorithm later $$ \begin{align} C_1 X_1 &= D_1\\ C_2 X_2 &= D_2 \end{align} $$
Due to some properties of $A$ and $B$, the first equation can be rewritten into $$ \begin{equation} \begin{bmatrix} \hat{A}&\hat{A}\\\hat{B}&-\hat{B} \end{bmatrix} \begin{bmatrix} X_1\\X_2 \end{bmatrix} = \begin{bmatrix} \hat{G}_1\\\hat{G}_2 \end{bmatrix} \end{equation} $$ Any of the two converted into the second form can be accepted, is this possible?
ps1:
The left matrix of the first equation is below $$ \begin{equation} \begin{bmatrix} J_1(kr)/r& k J_1'(kr) \\k J_1'(kr)&J_1(kr)/r \end{bmatrix} \begin{bmatrix} X_1\\X_2 \end{bmatrix} = \begin{bmatrix} G_1\\G_2 \end{bmatrix} \end{equation} $$
The problem is to find two matrices $D_1$ and $D_2$ that satisfies $$ D_1 J_1(kr)/r + D_2 kJ_1'(kr) = 0 $$ Maybe this involves some properties of Bessel function.
It can be rewritten in further: find function $x_1(r)$ and $x_2(r)$ that satisfies $$ \int k J_1'(kr) x_1(r)dr + \int J_1(kr) / r x_2(r)dr = 0 $$