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

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

Block Cholesky decomposition

Consider a block matrix:

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

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:

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

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

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

and the half matrices can be calculated by means of Cholesky decomposition or LDL decomposition. The half matrices satisfy that

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

Thus, we have

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

where

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

The matrix $\begin{matrix}LU\end{matrix}$ can be decomposed in an algebraic manner into

$L = \begin{pmatrix} A^{\frac{1}{2}} & 0 \\ C A^{-\frac{*}{2}} & Q^{\frac{1}{2}} \end{pmatrix} \mathrm{~~and~~} U = \begin{pmatrix} A^{\frac{*}{2}} & A^{-\frac{1}{2}}B \\ 0 & Q^{\frac{*}{2}} \end{pmatrix}.$