# Partial inverse of a matrix - or does it have its own name?

In my calculations I need to use something which is "between" a matrix and its inverse. That is, I invert only some dimensions. I am interested if it has an established name.

That is, a matrix (here 2x2 real, but it is more general) $$\begin{bmatrix} u' \\ v' \end{bmatrix} = M \begin{bmatrix} u \\ v \end{bmatrix}$$ defines a hyperplane in coordinates $(u,v,u',v')$. Its inverse (if exists) can be defined as a linear operator such that $$\begin{bmatrix} u \\ v \end{bmatrix} = M^{-1} \begin{bmatrix} u' \\ v' \end{bmatrix}.$$

I am interested in inverting only some coordinates, e.g. $$\begin{bmatrix} u \\ v' \end{bmatrix} = M^{(-1,1)} \begin{bmatrix} u' \\ v \end{bmatrix}.$$

I know it is a relatively simple thing related to the implicit function theorem, with simple formulas. Yet, I use it a lot and I need to call it somehow. So:

• does it have its own name?
• if not, is "partial inverse" fine? (not colliding with other names, not (too) confusing, etc)

If you are curious, I use it in physics (optics) to relate a scattering matrix (relating input to output) to a transfer matrix (relating left/right of an interface).

• It is used in optics, just with no name (as often it is used only once). My paper bases on jumping back and forth (e.g. unitary matrix -> partial inverse -> diagonalization -> partial inverse of $C$, $D$ and $C^{-1}$). In any case, all other names than partial inverse seem to me a bit contrived (i.e. there is no way I would have guessed what do they mean). – Piotr Migdal Nov 3 '14 at 13:31