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.[1]
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}.
See also
References
- ^ Gallivan, K. A.; Plemmons, R. J.; Sameh, A. H. (1990). "Parallel Algorithms for Dense Linear Algebra Computations". SIAM Review. 32 (1): 94–95. ISSN 0036-1445. Retrieved 24 June 2025.