Benutzer:Rfzn/Matrix decomposition

2021-01-31T18:30:13Z

42.117.252.235: /* Cholesky decomposition */ Little mistake need verified

{{Distinguish|matrix factorization of a polynomial}}
{{Short description|Representation of a matrix as a product}}
In the [[mathematics|mathematical]] discipline of [[linear algebra]], a '''matrix decomposition''' or '''matrix factorization''' is a [[factorization]] of a [[Matrix (mathematics)|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 [[algorithm]]s.

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]].

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'' = ''Q''T''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]].

== Decompositions related to solving systems of linear equations ==

=== LU decomposition ===
{{main|LU decomposition}}
*Applicable to: [[square matrix]] ''A''
*Decomposition: <math>A=LU</math>, where ''L'' is [[triangular matrix|lower triangular]] and ''U'' is [[triangular matrix|upper triangular]]
*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]].
*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]].
*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, then the LDU decomposition exists.<ref>{{harvnb|Simon|Blume|1994}} Chapter 7.</ref>
*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.

=== LU reduction ===
{{main|LU reduction}}

=== Block LU decomposition ===
{{main|Block LU decomposition}}

=== Rank factorization ===
{{main|Rank factorization}}
*Applicable to: ''m''-by-''n'' matrix ''A'' of rank ''r''
*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
*Comment: The rank factorization can be used to [[Moore–Penrose pseudoinverse#Rank decomposition|compute the Moore–Penrose pseudoinverse]] of ''A'',<ref>{{cite journal|last1=Piziak|first1=R.|last2=Odell|first2=P. L.|title=Full Rank Factorization of Matrices|journal=Mathematics Magazine|date=1 June 1999|volume=72|issue=3|pages=193|doi=10.2307/2690882|jstor=2690882}}</ref> which one can apply to [[Moore–Penrose pseudoinverse#Obtaining all solutions of a linear system|obtain all solutions of the linear system]] <math>Ax=b</math>.

=== Cholesky decomposition ===
{{main|Cholesky decomposition}}
*Applicable to: [[square matrix|square]], [[symmetric matrix|hermitian]], [[positive-definite matrix|positive definite]] matrix ''A''
*Decomposition: <math>A=U^*U^{T}</math>, where <math>U</math> is upper triangular with real positive diagonal entries
*Comment: if the matrix <math>A</math> is Hermitian and positive semi-definite, then it has a decomposition of the form <math>A=U^*U^{T}</math> if the diagonal entries of <math>U</math> 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 A is real and symmetric, <math>U</math> has all real elements
*Comment: An alternative is the [[LDL decomposition]], which can avoid extracting square roots.

=== QR decomposition ===
{{main|QR decomposition}}
*Applicable to: ''m''-by-''n'' matrix ''A'' with linearly independent columns
*Decomposition: <math>A=QR</math> where <math>Q</math> is a [[unitary matrix]] of size ''m''-by-''m'', and <math>R</math> is an [[triangular matrix|upper triangular]] matrix of size ''m''-by-''n''
*Uniqueness: In general it is not unique, but if <math>A</math> is of full [[Matrix rank|rank]], then there exists a single <math>R</math> that has all positive diagonal elements. If <math>A</math> is square, also <math>Q</math> is unique.
*Comment: The QR decomposition provides an alternative way of solving the system of equations <math>Ax=b</math> without [[matrix inverse|inverting]] the matrix <math>A</math>. The fact that <math>Q</math> is [[orthogonal matrix|orthogonal]] means that <math>Q^{\mathrm{T}}Q=I</math>, so that <math>Ax=b</math> is equivalent to <math>Rx=Q^{\mathrm{T}}b</math>, which is easier to solve since <math>R</math> is [[triangular matrix|triangular]].

=== RRQR factorization ===
{{main|RRQR factorization}}

=== Interpolative decomposition ===

{{main|Interpolative decomposition}}

== Decompositions based on eigenvalues and related concepts ==

=== Eigendecomposition ===
{{main|Eigendecomposition (matrix)}}
*Also called ''[[Spectral decomposition (Matrix)|spectral decomposition]]''.
*Applicable to: [[square matrix]] ''A'' with linearly independent eigenvectors (not necessarily distinct eigenvalues).
*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''.
*Existence: An ''n''-by-''n'' matrix ''A'' always has ''n'' (complex) 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>. <math>V</math> is invertible if and only if the ''n'' eigenvectors are [[Linear independence|linearly independent]] (i.e., each eigenvalue has [[geometric multiplicity]] equal to its [[algebraic 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: Every [[normal matrix]] ''A'' (i.e., matrix for which <math>AA^*=A^*A</math>, where <math>A^*</math> is a [[conjugate transpose]]) can be eigendecomposed. For a [[normal matrix]] ''A'' (and only for a normal matrix), the eigenvectors can also be made orthonormal (<math>VV^*=I</math>) and the eigendecomposition reads as <math>A=VDV^*</math>. In particular all [[Unitary matrix|unitary]], [[Hermitian matrix|Hermitian]], or [[Skew-Hermitian matrix|skew-Hermitian]] (in the real-valued case, all [[Orthogonal matrix|orthogonal]], [[Symmetric matrix|symmetric]], or [[Skew-symmetric matrix|skew-symmetric]], respectively) matrices are normal and therefore possess this property.
*Comment: For any real [[symmetric matrix]] ''A'', the eigendecomposition always exists and can be written as <math>A=VDV^T</math>, where both ''D'' and ''V'' 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 <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 understand than raising ''A'' to power ''t'', since ''A'' is usually not diagonal.

=== Jordan decomposition ===
The [[Jordan normal form]] and the [[Jordan–Chevalley decomposition]]
*Applicable to: [[square matrix]] ''A''
*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.

=== Schur decomposition ===
{{main|Schur decomposition}}
*Applicable to: [[square matrix]] ''A''
*Decomposition (complex version): <math>A=UTU^*</math>, where ''U'' is a [[unitary matrix]], <math>U^*</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.
*Comment: if A is a [[normal matrix]], then T is diagonal and the Schur decomposition coincides with the spectral decomposition.

=== Real Schur decomposition ===
*Applicable to: [[square matrix]] ''A''
*Decomposition: This is a version of Schur decomposition where <math>V</math> and <math>S</math> only contain real numbers. One can always write <math>A=VSV^T</math> where ''V'' is a real [[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).

=== QZ decomposition ===
{{main|QZ decomposition}}
*Also called: ''generalized Schur decomposition''
*Applicable to: [[square matrix|square matrices]] ''A'' and ''B''
*Comment: there are two versions of this decomposition: complex and real.
*Decomposition (complex version): <math>A=QSZ^*</math> and <math>B=QTZ^*</math> where ''Q'' and ''Z'' are [[unitary matrix|unitary matrices]], the * superscript represents [[conjugate transpose]], and ''S'' and ''T'' are [[upper triangular]] matrices.
*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).
*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.

=== Takagi's factorization ===
*Applicable to: square, complex, symmetric matrix ''A''.
*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''.
*Comment: The diagonal elements of ''D'' are the nonnegative square roots of the eigenvalues of <math>AA^*</math>.
*Comment: ''V'' may be complex even if ''A'' is real.
*Comment: This is not a special case of the eigendecomposition (see above), which uses <math>V^{-1}</math> instead of <math>V^T</math>. Moreover, if ''A'' is not real, it is not Hermitian and the form using <math>V^*</math> also does not apply.

=== Singular value decomposition ===
{{main|Singular value decomposition}}
*Applicable to: ''m''-by-''n'' matrix ''A''.
*Decomposition: <math>A=UDV^*</math>, where ''D'' is a nonnegative [[diagonal matrix]], and ''U'' and ''V'' satisfy <math>U^*U = I, V^*V = I</math>. Here <math>V^*</math> is the [[conjugate transpose]] of ''V'' (or simply the [[matrix transpose|transpose]], if ''V'' contains real numbers only), and ''I'' denotes the identity matrix (of some dimension).
*Comment: The diagonal elements of ''D'' are called the [[singular value]]s of ''A''.
*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 <math>A</math> are always uniquely determined. <math>U</math> and <math>V</math> need not to be unique in general.

=== Scale-invariant decompositions ===

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

*Applicable to: ''m''-by-''n'' matrix ''A''.
*Unit-Scale-Invariant Singular-Value Decomposition: <math>A=DUSV^*E</math>, where ''S'' is a unique nonnegative [[diagonal matrix]] of scale-invariant singular values, ''U'' and ''V'' are [[unitary matrix|unitary matrices]], <math>V^*</math> is the [[conjugate transpose]] of ''V'', and positive diagonal matrices ''D'' and ''E''.
*Comment: Is analogous to the SVD except that the diagonal elements of ''S'' are invariant with respect to left and/or right multiplication of ''A'' 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 of ''A'' by arbitrary unitary matrices.
*Comment: Is an alternative to the standard SVD when invariance is required with respect to diagonal rather than unitary transformations of ''A''.
*Uniqueness: The scale-invariant singular values of <math>A</math> (given by the diagonal elements of ''S'') are always uniquely determined. Diagonal matrices ''D'' and ''E'', and unitary ''U'' and ''V'', are not necessarily unique in general.
*Comment: ''U'' and ''V'' matrices are not the same as those from the SVD.

Analogous scale-invariant decompositions can be derived from other matrix decompositions, e.g., to obtain scale-invariant eigenvalues.<ref>{{citation|last=Uhlmann |first=J.K. |title=A Generalized Matrix Inverse that is Consistent with Respect to Diagonal Transformations |journal=SIAM Journal on Matrix Analysis |year=2018 |volume=239 |issue=2 |pages=781–800 |doi=10.1137/17M113890X }}</ref><ref>{{citation|last=Uhlmann |first=J.K. |title=A Rank-Preserving Generalized Matrix Inverse for Consistency with Respect to Similarity |journal=IEEE Control Systems Letters |issn=2475-1456 |year=2018 |doi=10.1109/LCSYS.2018.2854240 |arxiv=1804.07334 }}</ref>

== Other decompositions ==

=== Polar decomposition ===
{{main|Polar decomposition}}
*Applicable to: any square complex matrix ''A''.
*Decomposition: <math>A=UP</math> (right polar decomposition) or <math>A=P'U</math> (left polar decomposition), where ''U'' is a [[unitary matrix]] and ''P'' and ''P''' are [[positive semidefinite matrix|positive semidefinite]] [[Hermitian matrices]].
*Uniqueness: <math>P</math> is always unique and equal to <math>\sqrt{A^*A}</math> (which is always hermitian and positive semidefinite). If <math>A</math> is invertible, then <math>U</math> is unique.
*Comment: Since any Hermitian matrix admits a spectral decomposition with a unitary matrix, <math>P</math> can be written as <math>P=VDV^*</math>. Since <math>P</math> is positive semidefinite, all elements in <math>D</math> are non-negative. Since the product of two unitary matrices is unitary, taking <math>W=UV</math>one can write <math>A=U(VDV^*)=WDV^* </math> which is the singular value decomposition. Hence, the existence of the polar decomposition is equivalent to the existence of the singular value decomposition.

=== Algebraic polar decomposition ===
*Applicable to: square, complex, non-singular matrix ''A''.<ref>{{harvnb|Choudhury|Horn|1987|pp=219–225}}</ref>
*Decomposition: <math>A=QS</math>, where ''Q'' is a complex orthogonal matrix and ''S'' is complex symmetric matrix.
*Uniqueness: If <math>A^{T}A</math> has no negative real eigenvalues, then the decomposition is unique.<ref name=":0">{{Cite journal|last=Bhatia|first=Rajendra|date=2013-11-15|title=The bipolar decomposition|journal=Linear Algebra and Its Applications|volume=439|issue=10|pages=3031–3037|doi=10.1016/j.laa.2013.09.006|doi-access=free}}</ref>
*Comment: The existence of this decomposition is equivalent to <math>AA^{T}</math> being similar to <math>A^{T}A</math>.<ref>{{harvnb|Horn|merino|1995|pp=43–92}}</ref>
*Comment: A variant of this decomposition is <math>A=RC</math>, where ''R'' is a real matrix and ''C'' is a [[circular matrix]].<ref name=":0" />

=== Mostow's decomposition ===
{{main|Mostow decomposition}}
* Applicable to: square, complex, non-singular matrix ''A''.<ref>{{citation|last=Mostow|first= G. D.|title= Some new decomposition theorems for semi-simple groups|series= Mem. Amer. Math. Soc. |year=1955|volume=14|pages= 31–54|url=https://archive.org/details/liealgebrasandli029541mbp|publisher= American Mathematical Society}}</ref><ref>{{Cite book|title=Matrix Information Geometry|last=Nielsen|first=Frank|last2=Bhatia|first2=Rajendra|publisher=Springer|year=2012|isbn=9783642302329|pages=224|language=en|doi=10.1007/978-3-642-30232-9|arxiv = 1007.4402}}</ref>
* Decomposition: <math>A=Ue^{iM}e^{S}</math>, where ''U'' is unitary, ''M'' is real anti-symmetric and ''S'' is real symmetric.
* Comment: The matrix ''A'' can also be decomposed as <math>A=U_2e^{S_2}e^{iM_2}</math>, where ''U2'' is unitary, ''M2'' is real anti-symmetric and ''S2'' is real symmetric.<ref name=":0" />

=== Sinkhorn normal form ===
{{main|Sinkhorn's theorem}}
*Applicable to: square real matrix ''A'' with strictly positive elements.
*Decomposition: <math>A=D_{1}SD_{2}</math>, where ''S'' is [[Doubly stochastic matrix|doubly stochastic]] and ''D''1 and ''D''2 are real diagonal matrices with strictly positive elements.

=== Sectoral decomposition ===
*Applicable to: square, complex matrix ''A'' with [[numerical range]] contained in the sector <math>S_\alpha = \left\{r e^{i \theta} \in \mathbb{C} \mid r> 0, |\theta| \le \alpha < \frac{\pi}{2}\right\}</math>.
*Decomposition: <math>A = CZC^*</math>, where ''C'' is an invertible complex matrix and <math>Z = \operatorname{diag}\left(e^{i\theta_1},\ldots,e^{i\theta_n}\right)</math> with all <math>\left|\theta_j\right| \le \alpha </math>.<ref name=Zhang2014>{{cite journal|last1=Zhang|first1=Fuzhen|title=A matrix decomposition and its applications|journal=Linear and Multilinear Algebra|volume=63|issue=10|date=30 June 2014|pages=2033–2042|doi=10.1080/03081087.2014.933219|url=https://zenodo.org/record/851661/files/article.pdf}}</ref><ref>{{cite journal|last1=Drury|first1=S.W.|title=Fischer determinantal inequalities and Highamʼs Conjecture|journal=Linear Algebra and Its Applications|date=November 2013|volume=439|issue=10|pages=3129–3133|doi=10.1016/j.laa.2013.08.031|doi-access=free}}</ref>

=== Williamson's normal form ===
* Applicable to: square, [[Positive-definite matrix|positive-definite]] real matrix ''A'' with order 2''n''-by-2''n''.
* Decomposition: <math>A=S^{T}\textrm{diag}(D,D)S</math>, where <math>S \in \text{Sp}(2n)</math> is a [[symplectic matrix]] and ''D'' is a nonnegative ''n''-by-''n'' diagonal matrix.<ref>{{Cite journal|last=Idel|first=Martin|last2=Soto Gaona|first2=Sebastián|last3=Wolf|first3=Michael M.|date=2017-07-15|title=Perturbation bounds for Williamson's symplectic normal form|journal=Linear Algebra and Its Applications|volume=525|pages=45–58|doi=10.1016/j.laa.2017.03.013|arxiv=1609.01338}}</ref>

== Generalizations ==
{{Expand section|1=examples and additional citations|date=December 2014}}
There exist analogues of the SVD, QR, LU and Cholesky factorizations for '''quasimatrices''' and '''cmatrices''' or '''continuous matrices'''.<ref>{{harvnb|Townsend|Trefethen|2015}}</ref> 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 {{harvtxt|Fredholm|1903}}, {{harvtxt|Hilbert|1904}} and {{harvtxt|Schmidt|1907}}. For an account, and a translation to English of the seminal papers, see {{harvtxt|Stewart|2011}}.

== See also ==
* [[Matrix splitting]]
* [[Non-negative matrix factorization]]
* [[Principal component analysis]]

== Notes ==
{{reflist}}

== References ==
*{{cite journal|last1=Choudhury|first1=Dipa|last2=Horn|first2=Roger A.|title=A Complex Orthogonal-Symmetric Analog of the Polar Decomposition|journal=SIAM Journal on Algebraic and Discrete Methods|date=April 1987|volume=8|issue=2|pages=219–225|doi=10.1137/0608019}}
*{{citation|first=I.|last=Fredholm|title=Sur une classe d'´equations fonctionnelles|journal=Acta Mathematica|volume=27|pages=365–390|year=1903|language=fr|author-link=Ivar Fredholm|doi=10.1007/bf02421317|doi-access=free}}
*{{citation|first=D.|last=Hilbert|title=Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen|journal=Nachr. Königl. Ges. Gött|volume=1904|pages=49–91|year=1904|language=de|author-link=David Hilbert}}
*{{cite journal|last1=Horn|first1=Roger A.|last2=Merino|first2=Dennis I.|title=Contragredient equivalence: A canonical form and some applications|journal=Linear Algebra and Its Applications|date=January 1995|volume=214|pages=43–92|doi=10.1016/0024-3795(93)00056-6|doi-access=free}}
*{{Citation|last1=Meyer|first1=C. 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}}
*{{citation|first=E.|last=Schmidt|title=Zur Theorie der linearen und nichtlinearen Integralgleichungen. I Teil. Entwicklung willkürlichen Funktionen nach System vorgeschriebener|journal=Mathematische Annalen|volume=63|issue=4|pages=433–476|year=1907|language=de|author-link=Erhard Schmidt|doi=10.1007/bf01449770|url=https://zenodo.org/record/1428258}}
*{{Cite book|last=Simon|first=C.|last2=Blume|first2=L.|year=1994|title=Mathematics for Economists|publisher= Norton|isbn=978-0-393-95733-4}}
*{{citation|last=Stewart|first=G. W.|year=2011|title=Fredholm, Hilbert, Schmidt: three fundamental papers on integral equations|url=http://www.cs.umd.edu/~stewart/FHS.pdf|access-date=2015-01-06}}
*{{citation|last=Townsend|first=A.|last2=Trefethen|first2=L. N.|year=2015|title=Continuous analogues of matrix factorizations|journal=[[Proceedings of the Royal Society|Proc. R. Soc. A]]|volume=471|issue=2173|pages=20140585|doi=10.1098/rspa.2014.0585|pmid=25568618|pmc=4277194|bibcode=2014RSPSA.47140585T}}

== External links ==
*[http://www.bluebit.gr/matrix-calculator/ Online Matrix Calculator]
*[http://www.wolframalpha.com/input/?i=matrix+decomposition&rawformassumption={%22C%22,+%22matrix+decomposition%22}+-%3E+{%22Calculator%22}&rawformassumption={%22MC%22,%22%22}-%3E{%22Formula%22} Wolfram Alpha Matrix Decomposition Computation » LU and QR Decomposition]
*[http://eom.springer.de/M/m120140.htm Springer Encyclopaedia of Mathematics » Matrix factorization]
*[https://web.archive.org/web/20110314171151/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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Benutzer:Rfzn/Matrix decomposition