Editing Matrix decomposition (section)

=== 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]].
*Existence: An LUP decomposition exists for any square matrix ''A''. When ''P'' is an [[identity matrix]], the LUP decomposition reduces to the LU decomposition.
*Comments: The LUP and LU decompositions are useful in solving an ''n''-by-''n'' system of linear equations <math>A \mathbf{x} = \mathbf{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''&nbsp;=&nbsp;''I'', so an LU decomposition exists.