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 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: