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(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

Main article: LU decomposition

LU reduction

Main article: LU reduction

Block LU decomposition

Main article: Block LU decomposition

Rank factorization

Main article: Rank factorization

Cholesky decomposition

Main article: Cholesky decomposition

QR decomposition

Main article: QR decomposition

RRQR factorization

Main article: RRQR factorization

Interpolative decomposition

Main article: Interpolative decomposition

Decompositions based on eigenvalues and related concepts


Main article: Eigendecomposition (matrix)

Jordan decomposition

The Jordan normal form and the Jordan–Chevalley decomposition

Schur decomposition

Main article: Schur decomposition

Real Schur decomposition

QZ decomposition

Main article: QZ decomposition

Takagi's factorization

Singular value decomposition

Main article: 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.[3][4]

Hessenberg decomposition

Complete orthogonal decomposition

Main article: Complete orthogonal decomposition

Other decompositions

Polar decomposition

Main article: Polar decomposition

Algebraic polar decomposition

Mostow's decomposition

Sinkhorn normal form

Main article: Sinkhorn's theorem

Sectoral decomposition

Williamson's normal form

Matrix square root

Main article: Square root of a matrix


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).

See also



  1. ^ If a non-square matrix is used, however, then the matrix U will also have the same rectangular shape as the original matrix A. And so, calling the matrix U upper triangular would be incorrect as the correct term would be that U is the 'row echelon form' of A. Other than this, there are no differences in LU factorization for square and non-square matrices.


  1. ^ Lay, David C. (2016). Linear algebra and its applications. Steven R. Lay, Judith McDonald (Fifth Global ed.). Harlow. p. 142. ISBN 978-1-292-09223-2. OCLC 920463015.((cite book)): CS1 maint: location missing publisher (link)
  2. ^ Piziak, R.; Odell, P. L. (1 June 1999). "Full Rank Factorization of Matrices". Mathematics Magazine. 72 (3): 193. doi:10.2307/2690882. JSTOR 2690882.
  3. ^ Uhlmann, J.K. (2018), "A Generalized Matrix Inverse that is Consistent with Respect to Diagonal Transformations", SIAM Journal on Matrix Analysis and Applications, 239 (2): 781–800, doi:10.1137/17M113890X
  4. ^ Uhlmann, J.K. (2018), "A Rank-Preserving Generalized Matrix Inverse for Consistency with Respect to Similarity", IEEE Control Systems Letters, 3: 91–95, arXiv:1804.07334, doi:10.1109/LCSYS.2018.2854240, ISSN 2475-1456, S2CID 5031440
  5. ^ Choudhury & Horn 1987, pp. 219–225
  6. ^ a b c Bhatia, Rajendra (2013-11-15). "The bipolar decomposition". Linear Algebra and Its Applications. 439 (10): 3031–3037. doi:10.1016/j.laa.2013.09.006.
  7. ^ Horn & Merino 1995, pp. 43–92
  8. ^ Mostow, G. D. (1955), Some new decomposition theorems for semi-simple groups, Mem. Amer. Math. Soc., vol. 14, American Mathematical Society, pp. 31–54
  9. ^ Nielsen, Frank; Bhatia, Rajendra (2012). Matrix Information Geometry. Springer. p. 224. arXiv:1007.4402. doi:10.1007/978-3-642-30232-9. ISBN 9783642302329. S2CID 118466496.
  10. ^ Zhang, Fuzhen (30 June 2014). "A matrix decomposition and its applications". Linear and Multilinear Algebra. 63 (10): 2033–2042. doi:10.1080/03081087.2014.933219. S2CID 19437967.
  11. ^ Drury, S.W. (November 2013). "Fischer determinantal inequalities and Highamʼs Conjecture". Linear Algebra and Its Applications. 439 (10): 3129–3133. doi:10.1016/j.laa.2013.08.031.
  12. ^ Idel, Martin; Soto Gaona, Sebastián; Wolf, Michael M. (2017-07-15). "Perturbation bounds for Williamson's symplectic normal form". Linear Algebra and Its Applications. 525: 45–58. arXiv:1609.01338. doi:10.1016/j.laa.2017.03.013. S2CID 119578994.
  13. ^ Townsend & Trefethen 2015