Block LU decomposition
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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
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
Thus, we have
The matrix can be decomposed in an algebraic manner into