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).

up vote 7 down vote accepted

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

  • So, as I see, it has many names. Thanks for this review! – Piotr Migdal Nov 3 '14 at 9:06
  • @PiotrMigdal Yes, it's one of those operations that pop up unexpectedly in lots of different fields. It is interesting to find out that it has an application in optics, too! – Federico Poloni Nov 3 '14 at 13:20
  • 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
  • @PiotrMigdal Another one of its many names is exchange operator. Maybe you find that more enlightening? – Federico Poloni Nov 3 '14 at 14:02
  • 2
    beware of the other en.wikipedia.org/wiki/Exchange_operator – Carlo Beenakker Nov 3 '14 at 18:21

Partial Inversion For Linear Systems And Partial Closure Of Independence Graphs

  • Just curious, why inversion not inverse? – Piotr Migdal Nov 3 '14 at 8:59
  • 2
    the operation is called a partial inversion, the outcome a partial inverse – Carlo Beenakker Nov 3 '14 at 9:00

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