Partial inverse of a matrix - or does it have its own name?

Question

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).

Answer 1

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

Answer 2

Partial Inversion For Linear Systems And Partial Closure Of Independence Graphs

Answer 3

Partial inversion is a valid name, and yeah, it is still a hot topic in S-matrix theory and Pseudo-Unitary Quantum Mechanics.

The partial inversion transformation $\hat{Y}_{ik}$ over a general rectangular matrix $Z$, with entries $Z_{\rho\sigma}$, is basically the application of the entry-wise transformations below.

\begin{align} Z_{ik} &\to Z_{ik}^{-1} ; \\ Z_{i\sigma} &\to -Z_{ik}^{-1} Z_{i\sigma}, \quad \text{for all} \quad \sigma \neq k; \\ Z_{\rho k} &\to Z_{\rho k}Z_{ik}^{-1}, \quad \text{for all} \quad \rho \neq i; \\ Z_{\rho \sigma} &\to Z_{\rho \sigma} - Z_{\rho k} Z_{ik}^{-1} Z_{i\sigma}, \quad \text{for all} \quad \sigma \neq k, \rho \neq i. \end{align}

It has a diagrammatic representation in this paper similar to permutation paths.

If you use it as a matrix transformation that is simultaneous to the permutation of vector entries between the input and output of a $\mathbf{a}=M\mathbf{b}$ relation, then the equation system defined by this relation is conserved.

According to this paper, it can be used to:

1. reshape a linear equation system;
2. build a matrix representation for a renormalized diagram of a scattering process;
3. change pseudo-unitarity of a matrix, grow a quantum circuit, then come back to the original unitary group;
4. reversibly "invert" singular matrices (you can apply the regular matrix inversion via partial inversion for singular matrices if you consider special conventions for zero-handling, but it may lack basic properties of inverses);
5. define exact symbolic inverses for arbitrary dimensions of square matrices;
6. define a flexibilization of matrix inverse for arbitrary rectangular matrices;
7. define matrix inversion as a non-linear matrix operator that uses both (i) a commutator operator and (ii) a special commutator that resembles a Yang-Baxter equation;
8. define generalized inverses and their relations to generalized transpositions along (or around) any diagonal other than the main diagonal.

Also, according to the graphs in soaklot's answer to this other question, it can:

9. invert fully symbolic matrices on sympy (python package) for dimensions greater than (4,4) faster than sympy's default inversion algorithm.