Block LU decomposition

In linear algebra, a Block LU decomposition is a 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.

Consider a block matrix:

${\begin{pmatrix}A&B\\C&D\end{pmatrix}}={\begin{pmatrix}I\\CA^{-1}\end{pmatrix}}\,A\,{\begin{pmatrix}I&A^{-1}B\end{pmatrix}}+{\begin{pmatrix}0&0\\0&D-CA^{-1}B\end{pmatrix}},$

where the matrix $A$ is assumed to be non-singular.

We can also rewrite the above equation using the half matrices:

${\begin{pmatrix}A&B\\C&D\end{pmatrix}}={\begin{pmatrix}A^{\frac {1}{2}}\\CA^{-{\frac {1}{2}}}\end{pmatrix}}{\begin{pmatrix}A^{\frac {1}{2}}&A^{-{\frac {1}{2}}}B\end{pmatrix}}+{\begin{pmatrix}0&0\\0&Q^{\frac {1}{2}}\end{pmatrix}}{\begin{pmatrix}0&0\\0&Q^{\frac {1}{2}}\end{pmatrix}},$

where the Schur complement of ${\begin{matrix}A\end{matrix}}$ in the block matrix is defined by

${\begin{matrix}Q=D-CA^{-1}B\end{matrix}}$

and the half matrices can be calculated by means of singular value decomposition. The half matrices satisfy that

${\begin{matrix}A^{\frac {1}{2}}\,A^{-{\frac {1}{2}}}=I;\end{matrix}}\qquad {\begin{matrix}A^{\frac {1}{2}}\,A^{\frac {1}{2}}=A;\end{matrix}}\qquad {\begin{matrix}Q^{\frac {1}{2}}\,Q^{\frac {1}{2}}=Q.\end{matrix}}$

Thus, we have

${\begin{pmatrix}A&B\\C&D\end{pmatrix}}=LU,$

where

$LU={\begin{pmatrix}A^{\frac {1}{2}}&0\\CA^{-{\frac {1}{2}}}&0\end{pmatrix}}{\begin{pmatrix}A^{\frac {1}{2}}&A^{-{\frac {1}{2}}}B\\0&0\end{pmatrix}}+{\begin{pmatrix}0&0\\0&Q^{\frac {1}{2}}\end{pmatrix}}{\begin{pmatrix}0&0\\0&Q^{\frac {1}{2}}\end{pmatrix}};$

$L={\begin{pmatrix}A^{\frac {1}{2}}&0\\CA^{-{\frac {1}{2}}}&Q^{\frac {1}{2}}\end{pmatrix}};$

$U={\begin{pmatrix}A^{\frac {1}{2}}&A^{-{\frac {1}{2}}}B\\0&Q^{\frac {1}{2}}\end{pmatrix}}.$