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| In the [[mathematics|mathematical]] discipline of [[linear algebra]], a '''matrix decomposition''' or '''matrix factorization''' is a [[factorization]] of a [[matrix (math)|matrix]] into a product of matrices. There are many different matrix decompositions; each finds use among a particular class of problems.
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| == Example ==
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| In [[numerical analysis]], different decompositions are used to implement efficient matrix [[algorithm]]s.
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| For instance, when solving a [[system of linear equations]] <math>Ax=b</math>, 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 <math>L(Ux)=b</math> and <math>Ux=L^{-1}b</math> require fewer additions and multiplications to solve, compared with the original system <math>Ax=b</math>, though one might require significantly more digits in inexact arithmetic such as [[floating point]].
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| Similarly, the [[QR decomposition]] expresses ''A'' as ''QR'' with ''Q'' a [[unitary matrix]] and ''R'' an upper triangular matrix. The system ''Q''(''Rx'') = ''b'' is solved by ''Rx'' = ''Q''<sup>T</sup>''b'' = ''c'', and the system ''Rx'' = ''c'' is solved by '[[Triangular matrix#Forward and back substitution|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]].
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| == Decompositions related to solving systems of linear equations ==
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| === [[LU decomposition]] ===
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| *Applicable to: [[square matrix]] ''A''
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| *Decomposition: <math>A=LU</math>, where ''L'' is [[triangular matrix|lower triangular]] and ''U'' is [[triangular matrix|upper triangular]]
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| *Related: the [[LDU decomposition|''LDU'' decomposition]] is <math>A=LDU</math>, where ''L'' is [[triangular matrix|lower triangular]] with ones on the diagonal, ''U'' is [[triangular matrix|upper triangular]] with ones on the diagonal, and ''D'' is a [[diagonal matrix]].
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| *Related: the [[LUP decomposition|''LUP'' decomposition]] is <math>A=LUP</math>, where ''L'' is [[triangular matrix|lower triangular]], ''U'' is [[triangular matrix|upper triangular]], and ''P'' is a [[permutation matrix]].
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| *Existence: An LUP decomposition exists for any square matrix ''A''. When ''P'' is an [[identity matrix]], the LUP decomposition reduces to the LU decomposition. If the LU decomposition exists, the LDU decomposition does too.<ref>{{Cite book|author=C. Simon and L. Blume |year=1994|url=http://www.amazon.com/dp/0393957330?s9r=8a02b54114211f250114aa25ed201279&itemPosition=1 |title=Mathematics for Economists|chapter=Chapter 7|publisher= Norton|isbn= 0-393-95733-0}}</ref>
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| *Comments: The LUP and LU decompositions are useful in solving an ''n''-by-''n'' system of linear equations <math>Ax=b</math>. These decompositions summarize the process of [[Gaussian elimination]] in matrix form. Matrix ''P'' represents any row interchanges carried out in the process of Gaussian elimination. If Gaussian elimination produces the [[row echelon form]] without requiring any row interchanges, then ''P=I'', so an LU decomposition exists.
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| === [[LU reduction]] ===
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| === [[Block LU decomposition]] ===
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| === [[Rank factorization]] ===
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| *Applicable to: ''m''-by-''n'' matrix ''A'' of rank ''r''
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| *Decomposition: <math>A=CF</math> where ''C'' is an ''m''-by-''r'' full column rank matrix and ''F'' is an ''r''-by-''n'' full row rank matrix
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| *Comment: The rank factorization can be used to [[Moore–Penrose pseudoinverse#Rank decomposition|compute the Moore-Penrose pseudoinverse]] of ''A'', which one can applies to [[Moore-Penrose pseudoinverse#Obtaining all solutions of a linear system|obtain all solutions of the linear system]] <math>Ax=b</math>.
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| === [[Cholesky decomposition]] ===
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| *Applicable to: [[square matrix|square]], [[symmetric matrix|symmetric]], [[positive-definite matrix|positive definite]] matrix ''A''
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| *Decomposition: <math>A=U^TU</math>, where ''U'' is upper triangular with positive diagonal entries
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| *Comment: the Cholesky decomposition is unique
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| *Comment: the Cholesky decomposition is also applicable for complex [[Hermitian matrix|hermitian]] positive definite matrices
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| *Comment: An alternative is the [[LDL decomposition]] which can avoid extracting square roots.
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| === [[QR decomposition]] ===
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| *Applicable to: ''m''-by-''n'' matrix ''A''
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| *Decomposition: <math>A=QR</math> where ''Q'' is an [[orthogonal matrix]] of size ''m''-by-''m'', and ''R'' is an [[triangular matrix|upper triangular]] matrix of size ''m''-by-''n''
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| *Comment: The QR decomposition provides an alternative way of solving the system of equations <math>Ax=b</math> without [[matrix inverse|inverting]] the matrix ''A''. The fact that ''Q'' is [[orthogonal matrix|orthogonal]] means that <math>Q^TQ=I</math>, so that <math>Ax=b</math> is equivalent to <math>Rx=Q^Tb</math>, which is easier to solve since ''R'' is [[triangular matrix|triangular]].
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| === [[RRQR factorization]] ===
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| === [[Singular value decomposition]] ===
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| *Applicable to: ''m''-by-''n'' matrix ''A''.
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| *Decomposition: <math>A=UDV^H</math>, where ''D'' is a nonnegative [[diagonal matrix]], and ''U'' and ''V'' are [[unitary matrix|unitary matrices]], and <math>V^H</math> denotes the [[conjugate transpose]] of ''V'' (or simply the [[matrix transpose|transpose]], if ''V'' contains real numbers only).
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| *Comment: The diagonal elements of ''D'' are called the [[singular value]]s of ''A''.
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| *Comment: like the eigendecomposition below, 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.
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| == Decompositions based on eigenvalues and related concepts ==
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| === [[Eigendecomposition (matrix)|Eigendecomposition]] ===
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| *Also called ''spectral decomposition''
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| *Applicable to: [[square matrix]] ''A'' with distinct eigenvectors (not necessarily distinct eigenvalues).
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| *Decomposition: <math>A=VDV^{-1}</math>, where ''D'' is a [[diagonal matrix]] formed from the [[eigenvalue]]s of ''A'', and the columns of ''V'' are the corresponding [[eigenvector]]s of ''A''.
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| *Existence: An ''n''-by-''n'' matrix ''A'' always has ''n'' eigenvalues, which can be ordered (in more than one way) to form an ''n''-by-''n'' diagonal matrix ''D'' and a corresponding matrix of nonzero columns ''V'' that satisfies the [[Eigenvalue, eigenvector and eigenspace#Definitions: the eigenvalue equation|eigenvalue equation]] <math>AV=VD</math>. If the ''n'' eigenvectors (not necessarily eigenvalues) are distinct (that is, none is equal to any of the others), then ''V'' is invertible, implying the decomposition <math>A=VDV^{-1}</math>.<ref>{{Citation | last1=Meyer | first1=Carl D. | title=Matrix Analysis and Applied Linear Algebra | url=http://www.matrixanalysis.com/ | publisher=[[Society for Industrial and Applied Mathematics|SIAM]] | isbn=978-0-89871-454-8 | year=2000 | page=514}}.</ref>
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| *Comment: One can always normalize the eigenvectors to have length one (see definition of the eigenvalue equation). If <math>A</math> is symmetric, <math>V</math> is always invertible and can be made to have normalized columns. Then the equation <math>VV^T=I</math> holds, because each eigenvector is orthogonal to the other. Therefore the decomposition (which always exists if ''A'' is symmetric) reads as: <math>A=VDV^T</math>
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| *Comment: The condition of having ''n'' distinct eigenvalues is sufficient but not necessary. The necessary and sufficient condition is for each eigenvalue to have geometric multiplicity equal to its algebraic multiplicity.
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| *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 <math>x_{t+1}=Ax_t</math> starting from the initial condition <math>x_0=c</math> is solved by <math>x_t = A^tc</math>, which is equivalent to <math>x_t = VD^tV^{-1}c</math>, where ''V'' and ''D'' are the matrices formed from the eigenvectors and eigenvalues of ''A''. Since ''D'' is diagonal, raising it to power <math>D^t</math>, just involves raising each element on the diagonal to the power ''t''. This is much easier to do and to understand than raising ''A'' to power ''t'', since ''A'' is usually not diagonal.
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| === Jordan decomposition ===
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| The [[Jordan normal form]] and the [[Jordan–Chevalley decomposition]]
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| *Applicable to: [[square matrix]] ''A''
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| *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.
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| === [[Schur decomposition]] ===
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| *Applicable to: [[square matrix]] ''A''
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| *Comment: there are two versions of this decomposition: the complex Schur decomposition and the real Schur decomposition. A complex matrix always has a complex Schur decomposition. A real matrix admits a real Schur decomposition if and only if all of its eigenvalues are real.
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| *Decomposition (complex version): <math>A=UTU^H</math>, where ''U'' is a [[unitary matrix]], <math>U^H</math> is the [[conjugate transpose]] of ''U'', and ''T'' is an [[upper triangular]] matrix called the complex [[Schur form]] which has the [[eigenvalue]]s of ''A'' along its diagonal.
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| *Decomposition (real version): <math>A=VSV^T</math>, where ''A'', ''V'', ''S'' and <math>V^T</math> are matrices that contain real numbers only. In this case, ''V'' is an [[orthogonal matrix]], <math>V^T</math> is the [[matrix transpose|transpose]] of ''V'', and ''S'' is a [[block matrix|block upper triangular]] matrix called the real [[Schur form]]. The blocks on the diagonal of ''S'' are of size 1×1 (in which case they represent real eigenvalues) or 2×2 (in which case they are derived from [[complex conjugate]] eigenvalue pairs).
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| === [[QZ decomposition]] ===
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| *Also called: ''generalized Schur decomposition''
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| *Applicable to: [[square matrix|square matrices]] ''A'' and ''B''
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| *Comment: there are two versions of this decomposition: complex and real.
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| *Decomposition (complex version): <math>A=QSZ^H</math> and <math>B=QTZ^H</math> where ''Q'' and ''Z'' are [[unitary matrix|unitary matrices]], the ''H'' superscript represents [[conjugate transpose]], and ''S'' and ''T'' are [[upper triangular]] matrices.
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| *Comment: in the complex QZ decomposition, the ratios of the diagonal elements of ''S'' to the corresponding diagonal elements of ''T'', <math>\lambda_i = S_{ii}/T_{ii}</math>, are the generalized [[eigenvalue]]s that solve the [[Eigendecomposition of a matrix#Additional topics|generalized eigenvalue problem]] <math>Av=\lambda Bv</math> (where <math>\lambda</math> is an unknown scalar and ''v'' is an unknown nonzero vector).
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| *Decomposition (real version): <math>A=QSZ^T</math> and <math>B=QTZ^T</math> where ''A'', ''B'', ''Q'', ''Z'', ''S'', and ''T'' are matrices containing real numbers only. In this case ''Q'' and ''Z'' are [[orthogonal matrix|orthogonal matrices]], the ''T'' superscript represents [[matrix transpose|transposition]], and ''S'' and ''T'' are [[block matrix|block upper triangular]] matrices. The blocks on the diagonal of ''S'' and ''T'' are of size 1×1 or 2×2.
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| === Takagi's factorization === | |
| *Applicable to: square, complex, symmetric matrix ''A''.
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| *Decomposition: <math>A=VDV^T</math>, where ''D'' is a real nonnegative [[diagonal matrix]], and ''V'' is [[unitary matrix|unitary]]. <math>V^T</math> denotes the [[matrix transpose]] of ''V''.
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| *Comment: the diagonal elements of ''D'' are the nonnegative square roots of the eigenvalues of <math>AA^H</math>.
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| *Comment: ''V'' may be complex even if ''A'' is real.
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| *Comment: This is a special case of the eigendecomposition (see above)
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| == Other decompositions ==
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| * [[Polar decomposition]]
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| * [[Proper orthogonal decomposition]]
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| * [[Matrix decomposition into clans]]
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| ==See also==
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| * [[Matrix splitting]]
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| * [[Non-negative matrix factorization]]
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| == References ==
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| {{reflist}}
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| ==External links==
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| *[http://www.bluebit.gr/matrix-calculator/ Online Matrix Calculator]
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| *[http://eom.springer.de/M/m120140.htm Springer Encyclopaedia of Mathematics » Matrix factorization]
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| * [http://www.graphlab.ml.cmu.edu/pmf.html GraphLab] GraphLab collaborative filtering library, large scale parallel implementation of matrix decomposition methods (in C++) for multicore.
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| {{linear algebra}}
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| [[Category:Matrix theory]]
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| [[Category:Matrix decompositions]]
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