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, 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 and require fewer additions and multiplications to solve, compared with the original system, 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 = QTb** = c, and the system Rx = 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 related to solving systems of linear equations

LU decomposition

LU reduction

Block LU decomposition

Rank factorization

Cholesky decomposition

QR decomposition

RRQR factorization

Interpolative decomposition

Decompositions based on eigenvalues and related concepts

Eigendecomposition

Jordan decomposition

The Jordan normal form and the Jordan–Chevalley decomposition

Schur decomposition

Real Schur decomposition

QZ decomposition

Takagi's factorization

Singular value 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; for example, to obtain scale-invariant eigenvalues.

Hessenberg decomposition

Complete orthogonal decomposition

Other decompositions

Polar decomposition

Algebraic polar decomposition

Mostow's decomposition

Sinkhorn normal form

Sectoral decomposition

Williamson's normal form

Matrix square root

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.

Citations