Matrix decomposition: Difference between revisions

Revision as of 22:14, 20 February 2025

Latest revision as of 06:53, 18 July 2025

Line 14:

=== 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]].