View source for Partial inverse of a matrix

{{About|the partial inverse of a matrix|the partial inverse of a function|Inverse function#Partial inverses}}
In [[linear algebra]] and [[statistics]], the '''partial inverse''' of a [[Matrix (mathematics)|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 <math> n \times n </math> matrix <math> A</math> over a vector space <math>V</math> partitioned into blocks:

:<math> A = \begin{pmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{pmatrix} </math>

If <math>A_{11}</math> is invertible, then the partial inverse of <math>A</math> around the '''pivot block''' <math>A_{11}</math> is created by inverting <math>A_{11}</math>, putting the [[Schur complement]] <math>A / A_{11}</math> in place of <math>A_{22}</math>, and adjusting the off-diagonal elements accordingly:<ref name="ppt"/>

:<math> \operatorname{inv}_1 A = \begin{pmatrix} (A_{11})^{-1} & - (A_{11})^{-1} A_{12} \\ A_{21}  (A_{11})^{-1} & A_{22} - A_{21}  (A_{11})^{-1}A_{12} \end{pmatrix} </math>

Conceptually, partial inversion corresponds to a rotation<ref name="TT">[https://terrytao.wordpress.com/2015/10/07/sweeping-a-matrix-rotates-its-graph/ Sweeping a matrix rotates its graph], </ref> of the [[Graph_of_a_function|graph]] of the matrix <math> (X, AX) \in V \times V</math>, such that, for conformally-partitioned column matrices <math>(x_1, x_2)^T</math> and <math>(y_1, y_2)^T</math>:<ref name="ppt">Tsatsomeros, M. J. (2000). Principal pivot transforms: properties and applications. Linear Algebra and its Applications, 307 (1-3), 151–165.</ref>

:<math> A \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = \begin{pmatrix} y_1 \\ y_2 \end{pmatrix} \Leftrightarrow 
\operatorname{inv}_1(A) \begin{pmatrix} y_1 \\ x_2 \end{pmatrix} = \begin{pmatrix} x_1 \\ y_2 \end{pmatrix} </math>

As defined this way, this operator is its own inverse: <math> \operatorname{inv}_k(\operatorname{inv}_k(A)) = A </math>, and if the pivot block <math>A_{11}</math> is chosen to be the entire matrix, then the transform simply gives the matrix inverse <math>A^{-1}</math>. 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 <math>(\operatorname{inv}_k)^2 (A) = -A</math>. 

The transform is often presented as a pivot around a single non-zero element <math>a_{kk}</math>, in which case one has

:<math>
\left[ \operatorname{inv}_k (A) \right]_{ij} = \begin{cases} 
\frac{1}{a_{kk}} & i = j = k \\
-\frac{a_{kj}}{a_{kk}} & i = k, j \neq k \\
\frac{a_{ik}}{a_{kk}} & i \neq k, j = k \\
a_{ij} - \frac{a_{ik} a_{kj}}{a_{kk}} & i \neq k, j \neq k
\end{cases}
</math>

Partial inverses obey a number of nice properties:<ref name="Leeuw">[http://deleeuwpdx.net/pubfolders/pivot/pivot.pdf Exceedingly Simple Principal Pivot Transforms]</ref>
* inversions around different blocks commute, so larger pivots may be built up from sequences of smaller ones
* partial inversion preserves the space of symmetric matrices

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.<ref name="TT" /> 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.<ref name="Leeuw"/>

== References ==
{{reflist}}

[[Category:Matrix theory]]