Seeking Feedback on Robust Solution for Consistent but ill-conditioned Linear System with Real Symmetric Positive Semidefinite Matrices

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Let’s denote v as the nullvector of B: B*v = 0;
Then, (A+v*v')^{-1} = ((A+v*v')^{-1}*A)*A^{-1} = A^{-1} - (A+v*v')^{-1}*(v*v')*A^{-1}
where v*v’ constructs the nullspace of B which can’t traverse to the leftmost unless it commutes with A.