Find inverse of non symmetric matrix projected on subspace

Question

Suppose I have a matrix $M\in\Bbb C^{n\times n}$ and a projector $P\in\Bbb C^{n\times n} = \phi \phi^\dagger$ of rank 1 . I am looking for a way to compute numerically efficiently $X = P M^{-1}$. My feeling is that there is no need to compute the full inverse of $M$ only to project it on a lower subspace, but I could not find any helpful relations. Is there an expression for $X$ that avoids computing the full inverse of $M$?

Answer 1

You only need to compute the vector $$\psi:=(M^\dagger)^{-1}\phi=(M^{-1})^\dagger\phi.$$ Then, $$PM^{-1}=\phi\phi^\dagger M^{-1}=\phi\psi^\dagger.$$