Definition In linear algebra and statistics, the partial inverse of a matrix is an operation related to Gaussian elimination which has applications in numerical analysis, statistics and physics. It is also known by various authors as the principal pivot transform, or as the sweep, gyration, or exchange operator, represented by i n v k {\displaystyle \mathrm {inv} _{k}} if restricted to blocks along the main diagonal, or by Y ^ k ℓ {\displaystyle {\hat {Y}}_{k\ell }} if considering the general case of any arbitrary block from the matrix. Given an n × n {\displaystyle n\times n} matrix A {\displaystyle A} over a vector space V {\displaystyle V} partitioned into blocks:
A = ( A 11 A 12 A 21 A 22 ) {\displaystyle A={\begin{pmatrix}A_{11}&A_{12}\\A_{21}&A_{22}\end{pmatrix}}}
If A 11 {\displaystyle A_{11}} is invertible, then the partial inverse of A {\displaystyle A} around the pivot block A 11 {\displaystyle A_{11}} is created by inverting A 11 {\displaystyle A_{11}} , putting the Schur complement A / A 11 {\displaystyle A/A_{11}} in place of A 22 {\displaystyle A_{22}} , and adjusting the off-diagonal elements accordingly:
inv 1 A = ( ( A 11 ) − 1 − ( A 11 ) − 1 A 12 A 21 ( A 11 ) − 1 A 22 − A 21 ( A 11 ) − 1 A 12 ) {\displaystyle \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}}}
Conceptually, partial inversion corresponds to a rotation of the graph of the matrix ( X , A X ) ∈ V × V {\displaystyle (X,AX)\in V\times V} , such that, for conformally-partitioned column matrices ( x 1 , x 2 ) T {\displaystyle (x_{1},x_{2})^{T}} and ( y 1 , y 2 ) T {\displaystyle (y_{1},y_{2})^{T}} :
… excerpt ends here. Continue reading the full article.
