Block LU decomposition - Alchetron, The Free Social Encyclopedia

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

( A B C D ) = ( I 0 C A − 1 I ) ( A 0 0 D − C A − 1 B ) ( I A − 1 B 0 I )

Consider a block matrix:

( A B C D ) = ( I C A − 1 ) A ( I A − 1 B ) + ( 0 0 0 D − C A − 1 B ) ,

where the matrix A is assumed to be non-singular, I is an identity matrix with proper dimension, and 0 is a matrix whose elements are all zero.

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

( A B C D ) = ( A 1 2 C A − 1 2 ) ( A 1 2 A − 1 2 B ) + ( 0 0 0 Q 1 2 ) ( 0 0 0 Q 1 2 ) ,

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

Q = D − C A − 1 B

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

A 1 2 A 1 2 = A ; A 1 2 A − 1 2 = I ; A − 1 2 A ∗ 2 = I ; Q 1 2 Q 1 2 = Q .

Thus, we have

( A B C D ) = L U ,

where

L U = ( A 1 2 0 C A − 1 2 0 ) ( A 1 2 A − 1 2 B 0 0 ) + ( 0 0 0 Q 1 2 ) ( 0 0 0 Q 1 2 ) .

The matrix L U can be decomposed in an algebraic manner into

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