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.
An LU decomposition is LDU (Lower-Diagonal-Upper) decomposition may be performed if formula_1 is non-singular. Consider a block matrix:
This may be useful for inversion if also formula_3 (the Schur complement) is non-singular:
An equivalent UDL decomposition exists if formula_5 is non-singular:
This may be useful for inversion if formula_7 is non-singular:
Block Cholesky decomposition.
If the matrix is symmetric, then an alternative simplification is as follows:
where the matrix formula_10 is assumed to be non-singular,
formula_11 is an identity matrix with proper dimension, and formula_12 is a matrix whose elements are all zero.
We can also rewrite the above equation using the half matrices:
where the Schur complement of formula_10
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 formula_19 can be decomposed in an algebraic manner into