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Assume that I have a system given as $$ \begin{bmatrix} A& -B\\ 0 & C \end{bmatrix} \begin{bmatrix} A^{-1}& 0\\ 0 & I\\ \end{bmatrix} \begin{bmatrix} D & B\\ 0 & I\\ \end{bmatrix} \begin{bmatrix} q_1\\ q_2\\ \end{bmatrix} = \begin{bmatrix} b_1\\ b_2\\ \end{bmatrix} $$ which I would like to solve for the $q$ vector. If $A$ is a regular matrix, there is no problem . However, my problem is related to the cases where $A$ can become a singular matrix. In these singular cases, I was wondering if I could develop a projection matrix, say $T$, which I can explicitly separate $q_1$ into two spaces. Namely, one component for the particular solution that would result from the solution of the singular system such as $Ax=b$ (when $A$ is singular) and the other in the null space of $A$, since I can not find a solution to this problem directly, the problem will always be written with the null space vector or vectors augmented with the particular solution. I am confused about this projection matrix when $A$ is singular.

What I would like to perform at the end is to find a transformation like

$$ \begin{bmatrix} q_1\\ q_2\\ \end{bmatrix} = T \begin{bmatrix} \tilde{q}_1\\ \alpha \\ q_2\\ \end{bmatrix} $$ where $\alpha$ represents the coordinate related the null space and $\tilde{q}_1$ will be related to the coordinate space where I can use the pseudoinverse to find the particular solution.

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