Editing QR decomposition (section)

==Column pivoting==
Pivoted QR differs from ordinary Gram-Schmidt in that it takes the largest remaining column at the beginning of each new step—column pivoting—<ref>{{cite book |last1=Strang |first1=Gilbert |title=Linear Algebra and Learning from Data |date=2019 |publisher=Wellesley Cambridge Press |location=Wellesley |isbn=978-0-692-19638-0 |page=143 |edition=1st}}</ref> and thus introduces a [[permutation matrix]] ''P'':
:<math>AP = QR\quad \iff\quad A = QRP^\textsf{T}</math>

Column pivoting is useful when ''A'' is (nearly) [[rank deficient]], or is suspected of being so. It can also improve numerical accuracy. ''P'' is usually chosen so that the diagonal elements of ''R'' are non-increasing: <math>\left|r_{11}\right| \ge \left|r_{22}\right| \ge \cdots \ge \left|r_{nn}\right|</math>. This can be used to find the (numerical) rank of ''A'' at lower computational cost than a [[singular value decomposition]], forming the basis of so-called [[rank-revealing QR algorithm]]s.