Editing Perron–Frobenius theorem (section)

{{Short description|Theory in linear algebra}}
In [[matrix theory]], the '''Perron–Frobenius theorem''', proved by {{harvs|txt|authorlink=Oskar Perron|first=Oskar|last= Perron|year=1907}} and {{harvs|txt|authorlink=Georg Frobenius|first=Georg |last=Frobenius|year=1912}}, asserts that a [[real square matrix]] with positive entries has a unique [[eigenvalue]] of largest magnitude and that eigenvalue is real.  The corresponding [[eigenvector]] can be chosen to have strictly positive components, and also asserts a similar statement for certain classes of [[nonnegative matrices]]. This theorem has important applications to probability theory ([[ergodicity]] of [[Markov chain]]s); to the theory of [[dynamical systems]] ([[subshifts of finite type]]); to economics ([[Okishio's theorem]],<ref>{{Cite journal|last=Bowles|first=Samuel|date=1981-06-01|title=Technical change and the profit rate: a simple proof of the Okishio theorem|journal=Cambridge Journal of Economics|language=en|volume=5|issue=2|pages=183–186|doi=10.1093/oxfordjournals.cje.a035479|issn=0309-166X}}</ref> [[Hawkins–Simon condition]]<ref name="Meyer681">{{harvnb|Meyer|2000|pp=[http://www.matrixanalysis.com/Chapter8.pdf 8.3.6 p. 681] {{cite web |url=http://www.matrixanalysis.com/Chapter8.pdf |title=Archived copy |access-date=2010-03-07 |url-status=dead |archive-url=https://web.archive.org/web/20100307021652/http://www.matrixanalysis.com/Chapter8.pdf |archive-date=March 7, 2010 }}}}</ref>);
to demography ([[Leslie matrix|Leslie population age distribution model]]);<ref name="Meyer683">{{harvnb|Meyer|2000|pp=[http://www.matrixanalysis.com/Chapter8.pdf 8.3.7 p. 683] {{cite web |url=http://www.matrixanalysis.com/Chapter8.pdf |title=Archived copy |access-date=2010-03-07 |url-status=dead |archive-url=https://web.archive.org/web/20100307021652/http://www.matrixanalysis.com/Chapter8.pdf |archive-date=March 7, 2010 }}}}</ref> 
to social networks ([[DeGroot learning|DeGroot learning process]]); to  Internet search engines ([[PageRank]]);<ref name="LangvilleMeyer167">{{harvnb|Langville|Meyer|2006|p=[https://books.google.com/books?id=hxvB14-I0twC&pg=PA167 15.2 p. 167]}} {{cite book |url=https://books.google.com/books?id=hxvB14-I0twC&pg=PA167 |title=Google's PageRank and Beyond: The Science of Search Engine Rankings |access-date=2016-10-31 |url-status=bot: unknown |archive-url=https://web.archive.org/web/20140710041730/https://books.google.com/books?id=hxvB14-I0twC&lpg=PP1&dq=isbn%3A0691122024&pg=PA167 |archive-date=July 10, 2014 |isbn=978-0691122021 |last1=Langville |first1=Amy N.|author1-link= Amy Langville |last2=Langville |first2=Amy N. |last3=Meyer |first3=Carl D. |date=2006-07-23 |publisher=Princeton University Press }}</ref> and even to ranking of American football
teams.<ref name="Keener80">{{harvnb|Keener|1993|p=[https://www.jstor.org/stable/2132526 p. 80]}}</ref> The first to discuss the ordering of players within tournaments using Perron–Frobenius eigenvectors is [[Edmund Landau]].<ref>{{citation | title = Zur relativen Wertbemessung der Turnierresultaten| pages = 366–369| volume = XI |  journal = Deutsches Wochenschach | year=1895 | first1=Edmund | last1=Landau }}</ref><ref>{{citation | title = Über Preisverteilung bei Spielturnieren| pages = 192–202| volume = 63 |  journal =Zeitschrift für Mathematik und Physik | year=1915 | first1=Edmund | last1=Landau |  url = http://iris.univ-lille1.fr/handle/1908/2031 }}</ref>