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](http://en.wikipedia.org/wiki/Scattering-matrix_method) (relating input to output) to a [transfer matrix](http://en.wikipedia.org/wiki/Transfer-matrix_method_(optics)) (relating left/right of an interface).



Partial inverse of a matrix - or does it have its own name?
It is a [principal pivot transform](http://www.math.wsu.edu/faculty/tsat/files/t2.pdf), also known as *sweep operator* or *gyration*. You can check the linked review paper.


