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

An LU decomposition is LDU (Lower-Diagonal-Upper) decomposition may be performed if ${\textstyle A}$ is non-singular. Consider a block matrix:

{\begin{bmatrix}A&B\\C&D\end{bmatrix}}={\begin{bmatrix}I&0\\CA^{-1}&I\end{bmatrix}}{\begin{bmatrix}A&0\\0&D-CA^{-1}B\end{bmatrix}}{\begin{bmatrix}I&A^{-1}B\\0&I\end{bmatrix}}

This may be useful for inversion if also ${\textstyle D-CA^{-1}B}$ (the Schur complement) is non-singular:

{\begin{bmatrix}A&B\\C&D\end{bmatrix}}^{-1}={\begin{bmatrix}I&A^{-1}B\\0&I\end{bmatrix}}^{-1}{\begin{bmatrix}A&0\\0&D-CA^{-1}B\end{bmatrix}}^{-1}{\begin{bmatrix}I&0\\CA^{-1}&I\end{bmatrix}}^{-1}={\begin{bmatrix}I&-A^{-1}B\\0&I\end{bmatrix}}{\begin{bmatrix}A&0\\0&D-CA^{-1}B\end{bmatrix}}^{-1}{\begin{bmatrix}I&0\\-CA^{-1}&I\end{bmatrix}}

An equivalent UDL decomposition exists if ${\textstyle D}$ is non-singular:

{\begin{bmatrix}A&B\\C&D\end{bmatrix}}={\begin{bmatrix}I&BD^{-1}\\0&I\end{bmatrix}}{\begin{bmatrix}A-BD^{-1}C&0\\0&D\end{bmatrix}}{\begin{bmatrix}I&0\\D^{-1}C&I\end{bmatrix}}

This may be useful for inversion if ${\textstyle A-BD^{-1}C}$ is non-singular:

{\begin{bmatrix}A&B\\C&D\end{bmatrix}}^{-1}={\begin{bmatrix}I&0\\D^{-1}C&I\end{bmatrix}}^{-1}{\begin{bmatrix}A-BD^{-1}C&0\\0&D\end{bmatrix}}^{-1}{\begin{bmatrix}I&BD^{-1}\\0&I\end{bmatrix}}^{-1}={\begin{bmatrix}I&0\\-D^{-1}C&I\end{bmatrix}}{\begin{bmatrix}A-BD^{-1}C&0\\0&D\end{bmatrix}}^{-1}{\begin{bmatrix}I&-BD^{-1}\\0&I\end{bmatrix}}

Block Cholesky decomposition

If the matrix is symmetric, then an alternative simplification is as follows:

{\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 ${\begin{matrix}A\end{matrix}}$ is assumed to be non-singular, ${\begin{matrix}I\end{matrix}}$ is an identity matrix with proper dimension, and ${\begin{matrix}0\end{matrix}}$ is a matrix whose elements are all zero.

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