Matrix decomposition

In the mathematical discipline of linear algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices. There are many different matrix decompositions; each finds use among a particular class of problems.
Example.
In numerical analysis, different decompositions are used to implement efficient matrix algorithms.
For instance, when solving a system of linear equations formula_1, the matrix "A" can be decomposed via the LU decomposition. The LU decomposition factorizes a matrix into a lower triangular matrix "L" and an upper triangular matrix "U". The systems formula_2 and formula_3 require fewer additions and multiplications to solve, compared with the original system formula_1, though one might require significantly more digits in inexact arithmetic such as floating point.
Similarly, the QR decomposition expresses "A" as "QR" with "Q" an orthogonal matrix and "R" an upper triangular matrix. The system "Q"("Rx) = b is solved by "Rx = "Q"Tb = c, and the system "R"x = c is solved by 'back substitution'. The number of additions and multiplications required is about twice that of using the LU solver, but no more digits are required in inexact arithmetic because the QR decomposition is numerically stable.
Decompositions based on eigenvalues and related concepts.
Jordan decomposition.
The Jordan normal form and the Jordan–Chevalley decomposition
Scale-invariant decompositions.
Refers to variants of existing matrix decompositions, such as the SVD, that are invariant with respect to diagonal scaling.
Analogous scale-invariant decompositions can be derived from other matrix decompositions, e.g., to obtain scale-invariant eigenvalues.
Generalizations.
There exist analogues of the SVD, QR, LU and Cholesky factorizations for quasimatrices and cmatrices or continuous matrices. A ‘quasimatrix’ is, like a matrix, a rectangular scheme whose elements are indexed, but one discrete index is replaced by a continuous index. Likewise, a ‘cmatrix’, is continuous in both indices. As an example of a cmatrix, one can think of the kernel of an integral operator.
These factorizations are based on early work by , and . For an account, and a translation to English of the seminal papers, see .