Matrix decomposition: Difference between revisions

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In [[numerical analysis]], different decompositions are used to implement efficient matrix [[algorithm]]s.

For instanceexample, when solving a [[system of linear equations]] <math>A \mathbf{x} = \mathbf{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(U \mathbf{x}) = \mathbf{b}</math> and <math>U \mathbf{x} = L^{-1} \mathbf{b}</math> require fewer additions and multiplications to solve, compared with the original system <math>A \mathbf{x} = \mathbf{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''(''R'''''x''') = '''b''' is solved by ''R'''''x''' = ''Q''^T'''b''' = '''c''', and the system ''R'''''x''' = '''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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=== LU decomposition ===

{{main|LU decomposition}}

*Traditionally applicable to: [[square matrix]] ''A'', although rectangular matrices can be applicable.<ref>{{Cite book|last=Lay|first=David C.|url=https://www.worldcat.org/oclc/920463015|title=Linear algebra and its applications|date=2016|others=Steven R. Lay, Judith McDonald|isbn=978-1-292-09223-2|edition=Fifth Global|___location=Harlow|pages=142|oclc=920463015}}</ref><ref group="nb">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.</ref>

*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>PA=LU</math>, where ''L'' is [[triangular matrix|lower triangular]], ''U'' is [[triangular matrix|upper triangular]], and ''P'' is a [[permutation matrix]].

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*Applicable to: square, complex, symmetric matrix ''A''.

*Decomposition: <math>A=VDV^\mathsf{T}</math>, where ''D'' is a real nonnegative [[diagonal matrix]], and ''V'' is [[unitary matrix|unitary]]. <math>V^\mathsf{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^*=VD^2V^*{-1}</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^\mathsf{T}</math>. Moreover, if ''A'' is not real, it is not Hermitian and the form using <math>V^*</math> also does not apply.

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Analogous scale-invariant decompositions can be derived from other matrix decompositions; for example, 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 and Applications |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 |volume=3 |pages=91–95 |doi=10.1109/LCSYS.2018.2854240 |arxiv=1804.07334 |s2cid=5031440 }}</ref>

===Hessenberg decomposition===

*Applicable to: [[square matrix]] A.

*Decomposition: <math>A=PHP^*</math> where <math>H</math> is the [[Hessenberg matrix]] and <math>P</math> is a [[unitary matrix]].

*Comment: often the first step in the Schur decomposition.

===Complete orthogonal decomposition===

{{main|Complete orthogonal decomposition}}

*Also known as: ''UTV decomposition'', ''ULV decomposition'', ''URV decomposition''.

*Applicable to: ''m''-by-''n'' matrix ''A''.

*Decomposition: <math>A=UTV^*</math>, where ''T'' is a [[triangular matrix]], and ''U'' and ''V'' are [[unitary matrix|unitary matrices]].

*Comment: Similar to the singular value decomposition and to the Schur decomposition.

== Other decompositions ==

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*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^\mathsf{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^\mathsf{T}</math> being similar to <math>A^\mathsf{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" />