Editing Block LU decomposition

{{Unreferenced|date=December 2009}}
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==
:<math>
\begin{pmatrix}
 A & B \\
 C & D 
\end{pmatrix}
=
\begin{pmatrix}
I & 0 \\
C A^{-1} & I
\end{pmatrix}
\begin{pmatrix}
A & 0 \\
0 & D-C A^{-1} B
\end{pmatrix}
\begin{pmatrix}
I & A^{-1} B \\
0 & I
\end{pmatrix}
</math>

==Block Cholesky decomposition==
Consider a [[block matrix]]:
:<math>
\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},
</math>
where the matrix <math>\begin{matrix}A\end{matrix}</math> is assumed to be non-singular,
<math>\begin{matrix}I\end{matrix}</math> is an identity matrix with proper dimension, and <math>\begin{matrix}0\end{matrix}</math> is a matrix whose elements are all zero.

We can also rewrite the above equation using the half matrices:
:<math>
\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}
,</math>
where the [[Schur complement]] of <math>\begin{matrix}A\end{matrix}</math>
in the block matrix is defined by
:<math>
\begin{matrix}
Q = D - C A^{-1} B
\end{matrix}
</math> 
and the half matrices can be calculated by means of [[Cholesky decomposition]] or [[LDL decomposition]].
The half matrices satisfy that
:<math>
\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}</math>

Thus, we have
:<math>
\begin{pmatrix}
 A & B \\
 C & D 
\end{pmatrix}
=
LU,
</math>
where
:<math>
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}.
</math>

The matrix <math>\begin{matrix}LU\end{matrix}</math> can be decomposed in an algebraic manner into
::<math>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}.
</math>

==See also==
*[[Matrix decomposition]]

==References==
{{reflist}}

{{DEFAULTSORT:Block Lu Decomposition}}
[[Category:Matrix decompositions]]