consider $Ax=b$ in which $A$ is a symmetric matrix of size $m$ and rank $m-r$. On the other hand, $B_{r\times m}x=0$ provides $r$ equations by using which the system of equations can be reduced to a non-singular system of size $m-r$. Having all those matrices, is there any algorithm to do this automatically? I would strongly prefer to preserve symmetry of $A$.
Thanks,