Crout matrix decomposition
Encyclopedia
In linear algebra
, the Crout matrix decomposition is an LU decomposition
which decomposes a matrix
into a lower triangular matrix (L), an upper triangular matrix (U) and, although not always needed, a permutation matrix
(P).
The Crout matrix decomposition algorithm
differs slightly from the Doolittle method. Doolittle's method returns a unit lower triangular matrix and an upper triangular matrix, while the Crout method returns a lower triangular matrix and a unit upper triangular matrix.
So, if a matrix decomposition of a matrix A is such that:
being L a unit lower triangular matrix, D a diagonal matrix and U a unit upper triangular matrix, then Doolittle's method produces
and Crout's method produces
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...
, the Crout matrix decomposition is an LU decomposition
LU decomposition
In linear algebra, LU decomposition is a matrix decomposition which writes a matrix as the product of a lower triangular matrix and an upper triangular matrix. The product sometimes includes a permutation matrix as well. This decomposition is used in numerical analysis to solve systems of linear...
which decomposes a matrix
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
into a lower triangular matrix (L), an upper triangular matrix (U) and, although not always needed, a permutation matrix
Permutation matrix
In mathematics, in matrix theory, a permutation matrix is a square binary matrix that has exactly one entry 1 in each row and each column and 0s elsewhere...
(P).
The Crout matrix decomposition algorithm
Algorithm
In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...
differs slightly from the Doolittle method. Doolittle's method returns a unit lower triangular matrix and an upper triangular matrix, while the Crout method returns a lower triangular matrix and a unit upper triangular matrix.
So, if a matrix decomposition of a matrix A is such that:
- A = LDU
being L a unit lower triangular matrix, D a diagonal matrix and U a unit upper triangular matrix, then Doolittle's method produces
- A = L(DU)
and Crout's method produces
- A = (LD)U.