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

We can also rewrite the above equation using the half matrices:: $egin{pmatrix} A & B \ C & D end{pmatrix}=egin{pmatrix}A^{frac{1}{2 \C A^{-frac{*}{2end{pmatrix}egin{pmatrix}A^{frac{*}{2 & A^{-frac{1}{2Bend{pmatrix}+egin{pmatrix}0 & 0 \0 & Q^{frac{1}{2end{pmatrix}egin{pmatrix}0 & 0 \0 & Q^{frac{*}{2end{pmatrix},$ where the Schur complement of $egin{matrix}Aend{matrix}$ in the block matrix is defined by: $egin{matrix}Q = D - C A^{-1} Bend{matrix}$ and the half matrices can be calculated by means of Cholesky decomposition or LDL decomposition.The half matrices satisfy that: $egin{matrix}A^{frac{1}{2,A^{frac{*}{2=A;end{matrix}qquadegin{matrix}A^{frac{1}{2,A^{-frac{1}{2=I;end{matrix}qquadegin{matrix}A^{-frac{*}{2,A^{frac{*}{2=I;end{matrix}qquadegin{matrix}Q^{frac{1}{2,Q^{frac{*}{2=Q.end{matrix}$

Thus, we have: $egin{pmatrix} A & B \ C & D end{pmatrix}=LU,$ where: $LU =egin{pmatrix}A^{frac{1}{2 & 0 \C A^{-frac{*}{2 & 0end{pmatrix}egin{pmatrix}A^{frac{*}{2 & A^{-frac{1}{2B \0 & 0end{pmatrix}+egin{pmatrix}0 & 0 \0 & Q^{frac{1}{2end{pmatrix}egin{pmatrix}0 & 0 \0 & Q^{frac{*}{2end{pmatrix}.$

The matrix $egin{matrix}LUend{matrix}$ can be decomposed in an algebraic manner into:: $L = egin{pmatrix}A^{frac{1}{2 & 0 \C A^{-frac{*}{2 & Q^{frac{1}{2end{pmatrix}mathrm{~~and~~}U =egin{pmatrix}A^{frac{*}{2 & A^{-frac{1}{2B \0 & Q^{frac{*}{2end{pmatrix}.$

ee also

*Matrix decomposition

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