Editing Permutation matrix (section)

{{Short description|Matrix with exactly one 1 per row and column}}
{{Use American English|date = January 2019}}

In [[mathematics]], particularly in [[Matrix (mathematics)|matrix theory]], a '''permutation matrix''' is a square [[binary matrix]] that has exactly one entry of 1 in each row and each column with all other entries 0.<ref name="Artin Algebra">{{cite book |author1-link=Michael Artin |last1=Artin |first1=Michael |date=1991 |title=Algebra  |publisher=Prentice Hall |pages=24–26,118,259,322 |isbn=0-13-004763-5 |oclc=24364036}}</ref>{{rp|page=26}} An {{math|''n'' × ''n''}} permutation matrix can represent a [[permutation]] of {{mvar|n}} elements. Pre-[[matrix multiplication|multiplying]] an {{mvar|n}}-row matrix {{mvar|M}} by a permutation matrix {{mvar|P}}, forming {{mvar|PM}}, results in permuting the rows of {{mvar|M}}, while post-multiplying an {{mvar|n}}-column matrix {{mvar|M}}, forming {{mvar|MP}}, permutes the columns of {{mvar|M}}.

Every permutation matrix ''P'' is [[orthogonal matrix|orthogonal]], with its [[invertible matrix|inverse]] equal to its [[transpose]]: <math>P^{-1}=P^\mathsf{T}</math>.<ref name="Artin Algebra" />{{rp|page=26}}  Indeed, permutation matrices can be [[Characterization (mathematics)|characterized]] as the orthogonal matrices whose entries are all [[non-negative]].<ref>{{cite journal |last1=Zavlanos |first1=Michael M. |last2=Pappas |first2=George J. |date=November 2008 |title=A dynamical systems approach to weighted graph matching |url=https://www.sciencedirect.com/science/article/abs/pii/S0005109808002616 |journal=Automatica |volume=44 |issue=11 |pages=2817–2824 |doi=10.1016/j.automatica.2008.04.009 |s2cid=834305 |access-date=21 August 2022 |quote=Let <math>O_n</math> denote the set of <math>n \times n</math> orthogonal matrices and <math>N_n</math> denote the set of <math>n \times n</math> element-wise non-negative matrices. Then, <math>P_n = O_n \cap N_n</math>, where <math>P_n</math> is the set of <math>n \times n</math> permutation matrices.|citeseerx=10.1.1.128.6870 }}</ref>