Partial inverse of a matrix

In linear algebra and statistics, the partial inverse of a matrix is an operation related to Gaussian elimination which has applications in numerical analysis and statistics. It is also known by various authors as the principal pivot transform, or as the sweep, gyration, or exchange operator.
Given an formula_1 matrix formula_2 over a vector space formula_3 partitioned into blocks:
If formula_5 is invertible, then the partial inverse of formula_6 around the pivot block formula_5 is created by inverting formula_5, putting the Schur complement formula_9 in place of formula_10, and adjusting the off-diagonal elements accordingly:
Conceptually, partial inversion corresponds to a rotation of the graph of the matrix formula_12, such that, for conformally-partitioned column matrices formula_13 and formula_14:
As defined this way, this operator is its own inverse: formula_16, and if the pivot block formula_5 is chosen to be the entire matrix, then the transform simply gives the matrix inverse formula_18. Note that some authors define a related operation (under one of the other names) which is not an inverse per se; particularly, one common definition instead has formula_19. 
The transform is often presented as a pivot around a single non-zero element formula_20, in which case one has
Partial inverses obey a number of nice properties:
Use of the partial inverse in numerical analysis is due to the fact that there is some flexibility in the choices of pivots, allowing for non-invertible elements to be avoided, and because the operation of "rotation" (of the graph of the pivoted matrix) has better numerical stability than the "shearing" operation which is implicitly performed by Gaussian elimination. Use in statistics is due to the fact that the resulting matrix nicely decomposes into blocks which have useful meanings in the context of linear regression.