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).