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]].