Block LU decomposition
In linear algebra, a Block LU decomposition is a matrix decomposition of a block matrix into a lower block triangular matrix L and an upper block triangular matrix U. This decomposition is used in numerical analysis to reduce the complexity of the block matrix formula. == Block LDU decomposition == ( A B C D ) = ( I 0 C A − 1 I ) ( A 0 0 D − C A − 1 B ) ( I A − 1 B 0 I ) {\displaystyle {\begin{pmatrix}A&B\\C&D\end{pmatrix}}={\begin{pmatrix}I&0\\CA^{-1}&I\end{pmatrix}}{\begin{pmatrix}A&0\\0&D-CA^{-1}B\end{pmatrix}}{\begin{pmatrix}I&A^{-1}B\\0&I\end{pmatrix}}} == Block Cholesky decomposition == Consider a block matrix: ( A B C D ) = ( I C A − 1 ) A ( I A − 1 B ) + ( 0 0 0 D − C A − 1 B ) , {\displaystyle {\begin{pmatrix}A&B\\C&D\end{pmatrix}}={\begin{pmatrix}I\\CA^{-1}\end{pmatrix}}\,A\,{\begin{pmatrix}I&A^{-1}B\end{pmatrix}}+{\begin{pmatrix}0&0\\0&D-CA^{-1}B\end{pmatrix}},} where the matrix A {\displaystyle {\begin{matrix}A\end{matrix}}} is assumed to be non-singular, I {\displaystyle {\begin{matrix}I\end{matrix}}} is an identity matrix with proper dimension, and 0 {\displaystyle {\begin{matrix}0\end{matrix}}} is a matrix whose elements are all zero.