Editing Perron–Frobenius theorem (section)

==References==

*{{Citation | last1=Perron | first1=Oskar | title=Zur Theorie der Matrices | doi=10.1007/BF01449896 | year=1907 | journal= [[Mathematische Annalen]] | volume=64 | issue=2 | pages=248–263 | hdl=10338.dmlcz/104432 | s2cid=123460172 | author-link = Oskar Perron | url=http://gdz.sub.uni-goettingen.de/dms/load/toc/?PID=PPN235181684_0064 | hdl-access=free }}
*{{citation|first=Georg|last= Frobenius|title=Ueber Matrizen aus nicht negativen Elementen|journal= [[:wikisource:de:Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin|Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften]] |date=May 1912|pages=456–477 | author-link = Georg Frobenius }}
*{{citation|first=Georg|last= Frobenius|title=Über Matrizen aus positiven Elementen, 1 |journal= Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften   |year=1908|pages=471–476 | author-link = Georg Frobenius}}
*{{citation|first=Georg|last= Frobenius|title=Über Matrizen aus positiven Elementen, 2 |journal= Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften   |year=1909|pages=514–518 | author-link = Georg Frobenius }}
* {{citation |  title= The Theory of Matrices, Volume 2  | orig-year = 1959|  year= 2000 | first1=Felix | last1=Gantmacher | publisher = AMS Chelsea Publishing| author1-link=Felix Gantmacher| isbn= 978-0-8218-2664-5|
url = https://books.google.com/books?id=cyX32q8ZP5cC&q=preceding%20section&pg=PA53}} (1959 edition had different title: "Applications of the theory of matrices". Also the numeration of chapters is different in the two editions.)
*{{citation|title=Google page rank and beyond |year=2006 |first1=Amy |last1=Langville |first2=Carl |last2=Meyer |publisher=Princeton University Press |isbn=978-0-691-12202-1 |doi=10.1007/s10791-008-9063-y |s2cid=7646929 }}
*{{citation |  title=The Perron–Frobenius theorem and the ranking of football teams | year=1993 | first1=James | last1=Keener | jstor = 2132526| journal =  SIAM Review| volume = 35| number =1| pages = 80–93 | doi=10.1137/1035004
}}
*{{citation
 |title       = Matrix analysis and applied linear algebra
 |year        = 2000
 |first1      = Carl
 |last1       = Meyer
 |url         = http://www.matrixanalysis.com/Chapter8.pdf
 |publisher   = SIAM
 |isbn        = 978-0-89871-454-8
 |url-status     = dead
 |archive-url  = https://web.archive.org/web/20100307021652/http://www.matrixanalysis.com/Chapter8.pdf
 |archive-date = 2010-03-07
}}
*{{citation|first=Henryk |last= Minc |title=Nonnegative matrices |isbn=0-471-83966-3 |year=1988 |publisher=John Wiley&Sons,New York}}
*{{citation|first=V.|last= Romanovsky|title=Sur les zéros des matrices stocastiques
|journal= Bulletin de la Société Mathématique de France |year=1933|pages=213–219
| volume = 61 |doi= 10.24033/bsmf.1206|doi-access=free}}
*{{citation|first=Lothar |last= Collatz| author-link = Lothar Collatz|title=Einschließungssatz für die charakteristischen Zahlen von Matrizen
|journal= Mathematische Zeitschrift |year=1942|pages=221–226
|doi = 10.1007/BF01180013| volume = 48 | number =1  |s2cid= 120958677}}
*{{citation|first=Helmut|last= Wielandt|title=Unzerlegbare, nicht negative Matrizen
|journal= Mathematische Zeitschrift |year=1950|pages=642–648
|doi = 10.1007/BF02230720| volume = 52 | number =1  |hdl= 10338.dmlcz/100322|s2cid= 122189604|hdl-access=free}}

===Further reading===

* Abraham Berman, [[Robert J. Plemmons]], ''Nonnegative Matrices in the Mathematical Sciences'', 1994, SIAM. {{isbn|0-89871-321-8}}.
* [[Chris Godsil]] and [[Gordon Royle]], ''Algebraic Graph Theory'', Springer, 2001.
* A. Graham, ''Nonnegative Matrices and Applicable Topics in Linear Algebra'', John Wiley&Sons, New York, 1987.
* R. A. Horn and C.R. Johnson, ''Matrix Analysis'', Cambridge University Press, 1990
* Bas Lemmens and Roger Nussbaum, ''Nonlinear Perron-Frobenius Theory'', Cambridge Tracts in Mathematics 189, Cambridge Univ. Press, 2012.
* S. P. Meyn and R. L. Tweedie, [https://web.archive.org/web/20100619010320/https://netfiles.uiuc.edu/meyn/www/spm_files/book.html ''Markov Chains and Stochastic Stability''] London: Springer-Verlag, 1993. {{isbn|0-387-19832-6}} (2nd edition, Cambridge University Press, 2009)
* Seneta, E. ''Non-negative matrices and Markov chains''. 2nd rev. ed., 1981, XVI, 288 p., Softcover Springer Series in Statistics. (Originally published by Allen & Unwin Ltd., London, 1973) {{isbn|978-0-387-29765-1}}
*{{eom|id=P/p072350|first=D.A. |last=Suprunenko}}  (The claim that ''A''<sub>''j''</sub> has order ''n''/''h'' at the end of the statement of the theorem is incorrect.)
*{{citation|first=Richard S.|last=Varga|author-link=Richard S. Varga|title=Matrix Iterative Analysis|edition=2nd|publisher=Springer-Verlag|year=2002}}.

{{DEFAULTSORT:Perron-Frobenius Theorem}}
[[Category:Matrix theory]]
[[Category:Theorems in linear algebra]]
[[Category:Markov processes]]