Block LU decomposition
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.
\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}
:
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
and the half matrices can be calculated by means of Cholesky decomposition
or LDL decomposition.
The half matrices satisfy that
Thus, we have
where
The matrix
can be decomposed in an algebraic manner into
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
, a Block LU decomposition is a matrix decomposition
Matrix decomposition
In the mathematical discipline of linear algebra, a matrix decomposition is a factorization of a matrix into some canonical form. There are many different matrix decompositions; each finds use among a particular class of problems.- Example :...
of a block matrix
Block matrix
In the mathematical discipline of matrix theory, a block matrix or a partitioned matrix is a matrix broken into sections called blocks. Looking at it another way, the matrix is written in terms of smaller matrices. We group the rows and columns into adjacent 'bunches'. A partition is the rectangle...
into a lower block triangular matrix L and an upper block triangular matrix U. This decomposition is used in numerical analysis
Numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....
to reduce the complexity of the block matrix formula.
\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}
Block Cholesky decomposition
Consider a block matrixBlock matrix
In the mathematical discipline of matrix theory, a block matrix or a partitioned matrix is a matrix broken into sections called blocks. Looking at it another way, the matrix is written in terms of smaller matrices. We group the rows and columns into adjacent 'bunches'. A partition is the rectangle...
:
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
Schur complement
In linear algebra and the theory of matrices,the Schur complement of a matrix block is defined as follows.Suppose A, B, C, D are respectivelyp×p, p×q, q×p...
of
in the block matrix is defined by
and the half matrices can be calculated by means of Cholesky decomposition
Cholesky decomposition
In linear algebra, the Cholesky decomposition or Cholesky triangle is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose. It was discovered by André-Louis Cholesky for real matrices...
or LDL decomposition.
The half matrices satisfy that
Thus, we have
where
The matrix
can be decomposed in an algebraic manner into
