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.

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*(*R***x**) = **b** is solved by *R***x** = *Q*^{T}**b** = **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.

Main article: Polar decomposition |

- Applicable to: any square complex matrix
*A*. - Decomposition: (right polar decomposition) or (left polar decomposition), where
*U*is a unitary matrix and*P*and*P'*are positive semidefinite Hermitian matrices. - Uniqueness: is always unique and equal to (which is always hermitian and positive semidefinite). If is invertible, then is unique.
- Comment: Since any Hermitian matrix admits a spectral decomposition with a unitary matrix, can be written as . Since is positive semidefinite, all elements in are non-negative. Since the product of two unitary matrices is unitary, taking one can write which is the singular value decomposition. Hence, the existence of the polar decomposition is equivalent to the existence of the singular value decomposition.

- Applicable to: square, complex, non-singular matrix
*A*.^{[5]} - Decomposition: , where
*Q*is a complex orthogonal matrix and*S*is complex symmetric matrix. - Uniqueness: If has no negative real eigenvalues, then the decomposition is unique.
^{[6]} - Comment: The existence of this decomposition is equivalent to being similar to .
^{[7]} - Comment: A variant of this decomposition is , where
*R*is a real matrix and*C*is a circular matrix.^{[6]}

- Applicable to: square, complex, non-singular matrix
*A*.^{[8]}^{[9]} - Decomposition: , where
*U*is unitary,*M*is real anti-symmetric and*S*is real symmetric. - Comment: The matrix
*A*can also be decomposed as , where*U*_{2}is unitary,*M*_{2}is real anti-symmetric and*S*_{2}is real symmetric.^{[6]}

Main article: Sinkhorn's theorem |

- Applicable to: square real matrix
*A*with strictly positive elements. - Decomposition: , where
*S*is doubly stochastic and*D*_{1}and*D*_{2}are real diagonal matrices with strictly positive elements.

- Applicable to: square, complex matrix
*A*with numerical range contained in the sector . - Decomposition: , where
*C*is an invertible complex matrix and with all .^{[10]}^{[11]}

- Applicable to: square, positive-definite real matrix
*A*with order 2*n*×2*n*. - Decomposition: , where is a symplectic matrix and
*D*is a nonnegative*n*-by-*n*diagonal matrix.^{[12]}

Main article: Square root of a matrix |

- Decomposition: , not unique in general.
- In the case of positive semidefinite , there is a unique positive semidefinite such that .

This section needs expansion with: examples and additional citations. You can help by adding to it. (December 2014)

There exist analogues of the SVD, QR, LU and Cholesky factorizations for **quasimatrices** and **cmatrices** or **continuous matrices**.^{[13]} 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 Fredholm (1903), Hilbert (1904) and Schmidt (1907). For an account, and a translation to English of the seminal papers, see Stewart (2011).