TITLE: Partial inverse of a matrix - or does it have its own name?
QUESTION [8 upvotes]: 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).

REPLY [9 votes]: It is a principal pivot transform, also known as sweep operator or gyration. You can check the linked review paper.

REPLY [6 votes]: Partial Inversion For Linear Systems And Partial Closure Of Independence Graphs