Block LU decomposition

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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 \\ 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{1}{2}} & A^{-\frac{*}{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}.
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