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 Cholesky decomposition
Consider a block matrix:

where the matrix  is assumed to be non-singular,
 is an identity matrix with proper dimension, and  is a matrix whose elements are all zero.

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

where the Schur complement of 
in the block matrix is defined by

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

Thus, we have

where

The matrix  can be decomposed in an algebraic manner into

Block LDU decomposition
An alternative to LU decomposition is LDU (Lower-Diagonal-Upper) decomposition if  is non-singular, which may be simpler to implement:

This may be useful for inversion if also  is non-singular:

An equivalent UDL decomposition exists if  is non-singular:

This may be useful for inversion if  is non-singular: