I want to know how to solve this system of equations:
$$ \begin{bmatrix} R_{N} &-Q_{NM} \\ -Q_{NM}^T & P_M \end{bmatrix} \begin{bmatrix} a_N \\ b_M \end{bmatrix} = \begin{bmatrix} \epsilon^2\\ -\\ 0 \\ 0 \end{bmatrix} $$ The matrix is a 2 x 2 block matrix. $R_N$, $Q_{NM}$, $P_M$ are Toeplitz matrices. $N$ and $M$ represent the size of the dimensions of the matrices.
The solution will give vectors $a_N$ and $b_N$ and an error $\epsilon^2$.
What technique(s) can be used to solve this system? I don't understand the error term. The system also seems to be homogeneous which is also confusing me.
The equations are attempting to solve a weighted least squares problem. I don't know if I provided enough information, am happy to supply more, I just don't know what information may be most useful for someone to help me.