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.
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
The matrix can be decomposed in an algebraic manner into