Block LU decomposition

Block LU decomposition

In linear algebra, a Block LU decomposition is a 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:

begin{pmatrix}
A & B 
C & D
end{pmatrix} = begin{pmatrix} I C A^{-1} end{pmatrix} ,A, begin{pmatrix} I & A^{-1}B end{pmatrix} + begin{pmatrix} 0 & 0 0 & D-C A^{-1} B end{pmatrix}, where the matrix begin{matrix}Aend{matrix} is assumed to be non-singular, begin{matrix}Iend{matrix} is an identity matrix with proper dimension, and begin{matrix}0end{matrix} is a matrix whose elements are all zero.

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

begin{pmatrix}
A & B 
C & D
end{pmatrix} = begin{pmatrix} A^{frac{1}{2}} C A^{-frac{*}{2}} end{pmatrix} begin{pmatrix} A^{frac{*}{2}} & A^{-frac{1}{2}}B end{pmatrix} + begin{pmatrix} 0 & 0 0 & Q^{frac{1}{2}} end{pmatrix} begin{pmatrix} 0 & 0 0 & Q^{frac{*}{2}} end{pmatrix} , where the Schur complement of begin{matrix}Aend{matrix} in the block matrix is defined by
begin{matrix} Q = D - C A^{-1} B end{matrix} and the half matrices can be calculated by means of Cholesky decomposition or LDL decomposition. The half matrices satisfy that
begin{matrix} A^{frac{1}{2}},A^{frac{*}{2}}=A; end{matrix} qquad begin{matrix} A^{frac{1}{2}},A^{-frac{1}{2}}=I; end{matrix} qquad begin{matrix} A^{-frac{*}{2}},A^{frac{*}{2}}=I; end{matrix} qquad begin{matrix} Q^{frac{1}{2}},Q^{frac{*}{2}}=Q. end{matrix}

Thus, we have

begin{pmatrix}
A & B 
C & D
end{pmatrix} = LU, where
LU = begin{pmatrix} A^{frac{1}{2}} & 0 C A^{-frac{*}{2}} & 0 end{pmatrix} begin{pmatrix} A^{frac{*}{2}} & A^{-frac{1}{2}}B 0 & 0 end{pmatrix} + begin{pmatrix} 0 & 0 0 & Q^{frac{1}{2}} end{pmatrix} begin{pmatrix} 0 & 0 0 & Q^{frac{*}{2}} end{pmatrix}.

The matrix begin{matrix}LUend{matrix} can be decomposed in an algebraic manner into

L =
begin{pmatrix} A^{frac{1}{2}} & 0 C A^{-frac{*}{2}} & Q^{frac{1}{2}} end{pmatrix} mathrm{~~and~~} U = begin{pmatrix} A^{frac{*}{2}} & A^{-frac{1}{2}}B 0 & Q^{frac{*}{2}} end{pmatrix}.

See also

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