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 2×2 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).
Best Answer
It is a principal pivot transform, also known as sweep operator or gyration. You can check the linked review paper.