In themathematical discipline oflinear algebra, amatrix decomposition ormatrix factorization is afactorization of amatrix into a product of matrices. There are many different matrix decompositions; each finds use among a particular class of problems.

Innumerical analysis, different decompositions are used to implement efficient matrixalgorithms.

For example, when solving asystem of linear equations${\displaystyle A\mathbf{x}=\mathbf{b}}$, the matrixA can be decomposed via theLU decomposition. The LU decomposition factorizes a matrix into alower triangular matrixL and anupper triangular matrixU. The systems${\displaystyle L(U\mathbf{x})=\mathbf{b}}$ and${\displaystyle U\mathbf{x}=L^{-1}\mathbf{b}}$ require fewer additions and multiplications to solve, compared with the original system${\displaystyle A\mathbf{x}=\mathbf{b}}$, though one might require significantly more digits in inexact arithmetic such asfloating point.

Similarly, theQR decomposition expressesA asQR withQ anorthogonal matrix andR an upper triangular matrix. The systemQ(Rx) =b is solved byRx =Q^Tb =c, and the systemRx =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 isnumerically stable.

• Traditionally applicable to:square matrixA, although rectangular matrices can be applicable.^{[1][nb 1]}
• Decomposition:${\displaystyle A=LU}$, whereL islower triangular andU isupper triangular.
• Related: theLDU decomposition is${\displaystyle A=LDU}$, whereL islower triangular with ones on the diagonal,U isupper triangular with ones on the diagonal, andD is adiagonal matrix.
• Related: theLUP decomposition is${\displaystyle PA=LU}$, whereL islower triangular,U isupper triangular, andP is apermutation matrix.
• Existence: An LUP decomposition exists for any square matrixA. WhenP is anidentity matrix, the LUP decomposition reduces to the LU decomposition.
• Comments: The LUP and LU decompositions are useful in solving ann-by-n system of linear equations${\displaystyle A\mathbf{x}=\mathbf{b}}$. These decompositions summarize the process ofGaussian elimination in matrix form. MatrixP represents any row interchanges carried out in the process of Gaussian elimination. If Gaussian elimination produces therow echelon form without requiring any row interchanges, thenP = I, so an LU decomposition exists.

• Applicable to:m-by-n matrixA of rankr
• Decomposition:${\displaystyle A=CF}$ whereC is anm-by-r full column rank matrix andF is anr-by-n full row rank matrix
• Comment: The rank factorization can be used tocompute the Moore–Penrose pseudoinverse ofA,^[2] which one can apply toobtain all solutions of the linear system${\displaystyle A\mathbf{x}=\mathbf{b}}$.

• Applicable to:square,hermitian,positive definite matrix${\displaystyle A}$
• Decomposition:${\displaystyle A=U^{\ast }U}$, where${\displaystyle U}$ is upper triangular with real positive diagonal entries
• Comment: if the matrix${\displaystyle A}$ is Hermitian and positive semi-definite, then it has a decomposition of the form${\displaystyle A=U^{\ast }U}$ if the diagonal entries of${\displaystyle U}$ are allowed to be zero
• Uniqueness: for positive definite matrices Cholesky decomposition is unique. However, it is not unique in the positive semi-definite case.
• Comment: if${\displaystyle A}$ is real and symmetric,${\displaystyle U}$ has all real elements
• Comment: An alternative is theLDL decomposition, which can avoid extracting square roots.

• Applicable to:m-by-n matrixA with linearly independent columns
• Decomposition:${\displaystyle A=QR}$ where${\displaystyle Q}$ is aunitary matrix of sizem-by-m, and${\displaystyle R}$ is anupper triangular matrix of sizem-by-n
• Uniqueness: In general it is not unique, but if${\displaystyle A}$ is of fullrank, then there exists a single${\displaystyle R}$ that has all positive diagonal elements. If${\displaystyle A}$ is square, also${\displaystyle Q}$ is unique.
• Comment: The QR decomposition provides an effective way to solve the system of equations${\displaystyle A\mathbf{x}=\mathbf{b}}$. The fact that${\displaystyle Q}$ isorthogonal means that${\displaystyle Q^{\mathrm{T}}Q=I}$, so that${\displaystyle A\mathbf{x}=\mathbf{b}}$ is equivalent to${\displaystyle R\mathbf{x}=Q^{\mathsf{T}}\mathbf{b}}$, which is very easy to solve since${\displaystyle R}$ istriangular.

• Also calledspectral decomposition.
• Applicable to:square matrixA with linearly independent eigenvectors (not necessarily distinct eigenvalues).
• Decomposition:${\displaystyle A=VD{V}^{-1}}$, whereD is adiagonal matrix formed from theeigenvalues ofA, and the columns ofV are the correspondingeigenvectors ofA.
• Existence: Ann-by-n matrixA always hasn (complex) eigenvalues, which can be ordered (in more than one way) to form ann-by-n diagonal matrixD and a corresponding matrix of nonzero columnsV that satisfies theeigenvalue equation${\displaystyle AV=VD}$.${\displaystyle V}$ is invertible if and only if then eigenvectors arelinearly independent (that is, each eigenvalue hasgeometric multiplicity equal to itsalgebraic multiplicity). A sufficient (but not necessary) condition for this to happen is that all the eigenvalues are different (in this case geometric and algebraic multiplicity are equal to 1)
• Comment: One can always normalize the eigenvectors to have length one (see the definition of the eigenvalue equation)
• Comment: Everynormal matrixA (that is, matrix for which${\displaystyle A{A}^{\ast }=A^{\ast }A}$, where${\displaystyle A^{\ast }}$ is aconjugate transpose) can be eigendecomposed. For anormal matrixA (and only for a normal matrix), the eigenvectors can also be made orthonormal (${\displaystyle V{V}^{\ast }=I}$) and the eigendecomposition reads as${\displaystyle A=VD{V}^{\ast }}$. In particular allunitary,Hermitian, orskew-Hermitian (in the real-valued case, allorthogonal,symmetric, orskew-symmetric, respectively) matrices are normal and therefore possess this property.
• Comment: For any realsymmetric matrixA, the eigendecomposition always exists and can be written as${\displaystyle A=VD{V}^{\mathsf{T}}}$, where bothD andV are real-valued.
• Comment: The eigendecomposition is useful for understanding the solution of a system of linear ordinary differential equations or linear difference equations. For example, the difference equation${\displaystyle x_{t+1}=A{x}_t}$ starting from the initial condition${\displaystyle x_0=c}$ is solved by${\displaystyle x_t=A^{t}c}$, which is equivalent to${\displaystyle x_t=V{D}^t{V}^{-1}c}$, whereV andD are the matrices formed from the eigenvectors and eigenvalues ofA. SinceD is diagonal, raising it to power${\displaystyle D^t}$, just involves raising each element on the diagonal to the powert. This is much easier to do and understand than raisingA to powert, sinceA is usually not diagonal.

TheJordan normal form and theJordan–Chevalley decomposition

• Applicable to:square matrixA
• Comment: the Jordan normal form generalizes the eigendecomposition to cases where there are repeated eigenvalues and cannot be diagonalized, the Jordan–Chevalley decomposition does this without choosing a basis.

• Applicable to:square matrixA
• Decomposition (complex version):${\displaystyle A=UT{U}^{\ast }}$, whereU is aunitary matrix,${\displaystyle U^{\ast }}$ is theconjugate transpose ofU, andT is anupper triangular matrix called the complexSchur form which has theeigenvalues ofA along its diagonal.
• Comment: ifA is anormal matrix, thenT is diagonal and the Schur decomposition coincides with the spectral decomposition.

• Applicable to:square matrixA
• Decomposition: This is a version of Schur decomposition where${\displaystyle V}$ and${\displaystyle S}$ only contain real numbers. One can always write${\displaystyle A=VS{V}^{\mathsf{T}}}$ whereV is a realorthogonal matrix,${\displaystyle V^{\mathsf{T}}}$ is thetranspose ofV, andS is ablock upper triangular matrix called the realSchur form. The blocks on the diagonal ofS are of size 1×1 (in which case they represent real eigenvalues) or 2×2 (in which case they are derived fromcomplex conjugate eigenvalue pairs).

• Also called:generalized Schur decomposition
• Applicable to:square matricesA andB
• Comment: there are two versions of this decomposition: complex and real.
• Decomposition (complex version):${\displaystyle A=QS{Z}^{\ast }}$ and${\displaystyle B=QT{Z}^{\ast }}$ whereQ andZ areunitary matrices, the * superscript representsconjugate transpose, andS andT areupper triangular matrices.
• Comment: in the complex QZ decomposition, the ratios of the diagonal elements ofS to the corresponding diagonal elements ofT,${\displaystyle {\lambda }_i=S_{ii}/T_{ii}}$, are the generalizedeigenvalues that solve thegeneralized eigenvalue problem${\displaystyle A\mathbf{v}=\lambda B\mathbf{v}}$ (where${\displaystyle \lambda }$ is an unknown scalar andv is an unknown nonzero vector).
• Decomposition (real version):${\displaystyle A=QS{Z}^{\mathsf{T}}}$ and${\displaystyle B=QT{Z}^{\mathsf{T}}}$ whereA,B,Q,Z,S, andT are matrices containing real numbers only. In this caseQ andZ areorthogonal matrices, theT superscript representstransposition, andS andT areblock upper triangular matrices. The blocks on the diagonal ofS andT are of size 1×1 or 2×2.

• Applicable to: square, complex, symmetric matrixA.
• Decomposition:${\displaystyle A=VD{V}^{\mathsf{T}}}$, whereD is a real nonnegativediagonal matrix, andV isunitary.${\displaystyle V^{\mathsf{T}}}$ denotes thematrix transpose ofV.
• Comment: The diagonal elements ofD are the nonnegative square roots of the eigenvalues of${\displaystyle A{A}^{\ast }=V{D}^2{V}^{-1}}$.
• Comment:V may be complex even ifA is real.
• Comment: This is not a special case of the eigendecomposition (see above), which uses${\displaystyle V^{-1}}$ instead of${\displaystyle V^{\mathsf{T}}}$. Moreover, ifA is not real, it is not Hermitian and the form using${\displaystyle V^{\ast }}$ also does not apply.

• Applicable to:m-by-n matrixA.
• Decomposition:${\displaystyle A=UD{V}^{\ast }}$, whereD is a nonnegativediagonal matrix, andU andV satisfy${\displaystyle U^{\ast }U=I,V^{\ast }V=I}$. Here${\displaystyle V^{\ast }}$ is theconjugate transpose ofV (or simply thetranspose, ifV contains real numbers only), andI denotes the identity matrix (of some dimension).
• Comment: The diagonal elements ofD are called thesingular values ofA.
• Comment: Like the eigendecomposition above, the singular value decomposition involves finding basis directions along which matrix multiplication is equivalent to scalar multiplication, but it has greater generality since the matrix under consideration need not be square.
• Uniqueness: the singular values of${\displaystyle A}$ are always uniquely determined.${\displaystyle U}$ and${\displaystyle V}$ need not to be unique in general.

Refers to variants of existing matrix decompositions, such as the SVD, that are invariant with respect to diagonal scaling.

• Applicable to:m-by-n matrixA.
• Unit-Scale-Invariant Singular-Value Decomposition:${\displaystyle A=DUS{V}^{\ast }E}$, whereS is a unique nonnegativediagonal matrix of scale-invariant singular values,U andV areunitary matrices,${\displaystyle V^{\ast }}$ is theconjugate transpose ofV, and positive diagonal matricesD andE.
• Comment: Is analogous to the SVD except that the diagonal elements ofS are invariant with respect to left and/or right multiplication ofA by arbitrary nonsingular diagonal matrices, as opposed to the standard SVD for which the singular values are invariant with respect to left and/or right multiplication ofA by arbitrary unitary matrices.
• Comment: Is an alternative to the standard SVD when invariance is required with respect to diagonal rather than unitary transformations ofA.
• Uniqueness: The scale-invariant singular values of${\displaystyle A}$ (given by the diagonal elements ofS) are always uniquely determined. Diagonal matricesD andE, and unitaryU andV, are not necessarily unique in general.
• Comment:U andV matrices are not the same as those from the SVD.

Analogous scale-invariant decompositions can be derived from other matrix decompositions; for example, to obtain scale-invariant eigenvalues.^[3][4]

• Applicable to:square matrix A.
• Decomposition:${\displaystyle A=PH{P}^{\ast }}$ where${\displaystyle H}$ is theHessenberg matrix and${\displaystyle P}$ is aunitary matrix.
• Comment: often the first step in the Schur decomposition.

• Also known as:UTV decomposition,ULV decomposition,URV decomposition.
• Applicable to:m-by-n matrixA.
• Decomposition:${\displaystyle A=UT{V}^{\ast }}$, whereT is atriangular matrix, andU andV areunitary matrices.
• Comment: Similar to the singular value decomposition and to the Schur decomposition.

• Applicable to: any square complex matrixA.
• Decomposition:${\displaystyle A=UP}$ (right polar decomposition) or${\displaystyle A=P^{\prime }U}$ (left polar decomposition), whereU is aunitary matrix andP andP' arepositive semidefiniteHermitian matrices.
• Uniqueness:${\displaystyle P}$ is always unique and equal to${\displaystyle \sqrt{A^{\ast }A}}$ (which is always hermitian and positive semidefinite). If${\displaystyle A}$ is invertible, then${\displaystyle U}$ is unique.
• Comment: Since any Hermitian matrix admits a spectral decomposition with a unitary matrix,${\displaystyle P}$ can be written as${\displaystyle P=VD{V}^{\ast }}$. Since${\displaystyle P}$ is positive semidefinite, all elements in${\displaystyle D}$ are non-negative. Since the product of two unitary matrices is unitary, taking${\displaystyle W=UV}$one can write${\displaystyle A=U(VD{V}^{\ast })=WD{V}^{\ast }}$ 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 matrixA.^[5]
• Decomposition:${\displaystyle A=QS}$, whereQ is a complex orthogonal matrix andS is complex symmetric matrix.
• Uniqueness: If${\displaystyle A^{\mathsf{T}}A}$ has no negative real eigenvalues, then the decomposition is unique.^[6]
• Comment: The existence of this decomposition is equivalent to${\displaystyle A{A}^{\mathsf{T}}}$ being similar to${\displaystyle A^{\mathsf{T}}A}$.^[7]
• Comment: A variant of this decomposition is${\displaystyle A=RC}$, whereR is a real matrix andC is acircular matrix.^[6]

• Applicable to: square, complex, non-singular matrixA.^[8][9]
• Decomposition:${\displaystyle A=U{e}^{iM}{e}^S}$, whereU is unitary,M is real anti-symmetric andS is real symmetric.
• Comment: The matrixA can also be decomposed as${\displaystyle A=U_2{e}^{S_2}{e}^{i{M}_2}}$, whereU₂ is unitary,M₂ is real anti-symmetric andS₂ is real symmetric.^[6]

• Applicable to: square real matrixA with strictly positive elements.
• Decomposition:${\displaystyle A=D_{1}S{D}_2}$, whereS isdoubly stochastic andD₁ andD₂ are real diagonal matrices with strictly positive elements.

• Applicable to: square, complex matrixA withnumerical range contained in the sector.
• Decomposition:${\displaystyle A=CZ{C}^{\ast }}$, whereC is an invertible complex matrix and${\displaystyle Z=\mathrm{diag}\left(e^{i{\theta }_1},\dots ,e^{i{\theta }_n}\right)}$ with all${\displaystyle \left|{\theta }_j\right|\le \alpha }$.^[10][11]

• Applicable to: square,positive-definite real matrixA with order 2n×2n.
• Decomposition:${\displaystyle A=S^{\mathsf{T}}\mathrm{diag}(D,D)S}$, where${\displaystyle S\in \text{Sp}(2n)}$ is asymplectic matrix andD is a nonnegativen-by-n diagonal matrix.^[12]

• Decomposition:${\displaystyle A=BB}$, not unique in general.
• In the case of positive semidefinite${\displaystyle A}$, there is a unique positive semidefinite${\displaystyle B}$ such that${\displaystyle A=B^{\ast }B=BB}$.

This sectionneeds expansion with: examples and additional citations. You can help byadding missing information.(December 2014)

There exist analogues of the SVD, QR, LU and Cholesky factorizations forquasimatrices andcmatrices orcontinuous 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 anintegral operator.

These factorizations are based on early work byFredholm (1903),Hilbert (1904), andSchmidt (1907). For an account and a translation to English of the seminal papers, seeStewart (2011).

• Matrix splitting
• Non-negative matrix factorization
• Principal component analysis

• Choudhury, Dipa; Horn, Roger A. (April 1987). "A Complex Orthogonal-Symmetric Analog of the Polar Decomposition".SIAM Journal on Algebraic and Discrete Methods.8 (2):219–225.doi:10.1137/0608019.
• Fredholm, I. (1903), "Sur une classe d'´equations fonctionnelles",Acta Mathematica (in French),27:365–390,doi:10.1007/bf02421317
• Hilbert, D. (1904), "Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen",Nachr. Königl. Ges. Gött (in German),1904:49–91
• Horn, Roger A.; Merino, Dennis I. (January 1995)."Contragredient equivalence: A canonical form and some applications".Linear Algebra and Its Applications.214:43–92.doi:10.1016/0024-3795(93)00056-6.
• Meyer, C. D. (2000),Matrix Analysis and Applied Linear Algebra,SIAM,ISBN 978-0-89871-454-8
• Schmidt, E. (1907),"Zur Theorie der linearen und nichtlinearen Integralgleichungen. I Teil. Entwicklung willkürlichen Funktionen nach System vorgeschriebener",Mathematische Annalen (in German),63 (4):433–476,doi:10.1007/bf01449770
• Simon, C.; Blume, L. (1994).Mathematics for Economists. Norton.ISBN 978-0-393-95733-4.
• Stewart, G. W. (2011),Fredholm, Hilbert, Schmidt: three fundamental papers on integral equations(PDF), retrieved2015-01-06
• Townsend, A.; Trefethen, L. N. (2015), "Continuous analogues of matrix factorizations",Proc. R. Soc. A,471 (2173) 20140585,Bibcode:2014RSPSA.47140585T,doi:10.1098/rspa.2014.0585,PMC 4277194,PMID 25568618
• Jun, Lu (2021),Numerical matrix decomposition and its modern applications: A rigorous first course,arXiv:2107.02579

• Online Matrix CalculatorArchived 2008-12-12 at theWayback Machine
• Wolfram Alpha Matrix Decomposition Computation » LU and QR Decomposition
• Springer Encyclopaedia of Mathematics » Matrix factorization
• GraphLabGraphLabcollaborative filtering library, large scale parallel implementation of matrix decomposition methods (in C++) for multicore.

• Matrix theory
• Matrix decompositions
• Factorization

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