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Let a real square <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072350/p0723501.png" />-matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072350/p0723502.png" /> be considered as an operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072350/p0723503.png" />, let it be without invariant coordinate subspaces (such a matrix is called indecomposable) and let it be non-negative (i.e. all its elements are non-negative). Also, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072350/p0723504.png" /> be its eigen values, enumerated such that
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072350/p0723505.png" /></td> </tr></table>
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Let a real square  $  ( n \times n) $-
 +
matrix  $  A $
 +
be considered as an operator on  $  \mathbf R  ^ {n} $,
 +
let it be without invariant coordinate subspaces (such a matrix is called indecomposable) and let it be non-negative (i.e. all its elements are non-negative). Also, let  $  \lambda _ {1} \dots \lambda _ {n} $
 +
be its eigen values, enumerated such that
 +
 
 +
$$
 +
| \lambda _ {1} |  = \dots =  | \lambda _ {h} |  > | \lambda _ {h+} 1 |  \geq  \dots
 +
\geq  | \lambda _ {n} | ,\ \
 +
1 \leq  h \leq  n.
 +
$$
  
 
Then,
 
Then,
  
1) the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072350/p0723506.png" /> is a simple positive root of the [[Characteristic polynomial|characteristic polynomial]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072350/p0723507.png" />;
+
1) the number $  r = | \lambda _ {1} | $
 +
is a simple positive root of the [[Characteristic polynomial|characteristic polynomial]] of $  A $;
  
2) there exists an eigen vector of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072350/p0723508.png" /> with positive coordinates corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072350/p0723509.png" />;
+
2) there exists an eigen vector of $  A $
 +
with positive coordinates corresponding to $  r $;
  
3) the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072350/p07235010.png" /> coincide, apart from their numbering, with the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072350/p07235011.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072350/p07235012.png" />;
+
3) the numbers $  \lambda _ {1} \dots \lambda _ {h} $
 +
coincide, apart from their numbering, with the numbers $  r, \omega r \dots \omega  ^ {h-} 1 r $,  
 +
where $  \omega = e ^ {2 \pi i/h } $;
  
4) the product of any eigen value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072350/p07235013.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072350/p07235014.png" /> is an eigen value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072350/p07235015.png" />;
+
4) the product of any eigen value of $  A $
 +
by $  \omega $
 +
is an eigen value of $  A $;
  
5) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072350/p07235016.png" /> one can find a permutation of the rows and columns that reduces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072350/p07235017.png" /> to the form
+
5) for $  h > 1 $
 +
one can find a permutation of the rows and columns that reduces $  A $
 +
to the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072350/p07235018.png" /></td> </tr></table>
+
$$
 +
\left \|
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072350/p07235019.png" /> is a matrix of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072350/p07235020.png" />.
+
where $  A _ {j} $
 +
is a matrix of order $  nh  ^ {-} 1 $.
  
 
O. Perron proved the assertions 1) and 2) for positive matrices in [[#References|[1]]], while G. Frobenius [[#References|[2]]] gave the full form of the theorem.
 
O. Perron proved the assertions 1) and 2) for positive matrices in [[#References|[1]]], while G. Frobenius [[#References|[2]]] gave the full form of the theorem.
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  O. Perron,  "Zur Theorie der Matrizen"  ''Math. Ann.'' , '''64'''  (1907)  pp. 248–263</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G. Frobenius,  "Ueber Matrizen aus nicht negativen Elementen"  ''Sitzungsber. Königl. Preuss. Akad. Wiss.''  (1912)  pp. 456–477</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  F.R. [F.R. Gantmakher] Gantmacher,  "The theory of matrices" , '''1''' , Chelsea, reprint  (1977)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  O. Perron,  "Zur Theorie der Matrizen"  ''Math. Ann.'' , '''64'''  (1907)  pp. 248–263</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G. Frobenius,  "Ueber Matrizen aus nicht negativen Elementen"  ''Sitzungsber. Königl. Preuss. Akad. Wiss.''  (1912)  pp. 456–477</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  F.R. [F.R. Gantmakher] Gantmacher,  "The theory of matrices" , '''1''' , Chelsea, reprint  (1977)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Revision as of 08:05, 6 June 2020


Let a real square $ ( n \times n) $- matrix $ A $ be considered as an operator on $ \mathbf R ^ {n} $, let it be without invariant coordinate subspaces (such a matrix is called indecomposable) and let it be non-negative (i.e. all its elements are non-negative). Also, let $ \lambda _ {1} \dots \lambda _ {n} $ be its eigen values, enumerated such that

$$ | \lambda _ {1} | = \dots = | \lambda _ {h} | > | \lambda _ {h+} 1 | \geq \dots \geq | \lambda _ {n} | ,\ \ 1 \leq h \leq n. $$

Then,

1) the number $ r = | \lambda _ {1} | $ is a simple positive root of the characteristic polynomial of $ A $;

2) there exists an eigen vector of $ A $ with positive coordinates corresponding to $ r $;

3) the numbers $ \lambda _ {1} \dots \lambda _ {h} $ coincide, apart from their numbering, with the numbers $ r, \omega r \dots \omega ^ {h-} 1 r $, where $ \omega = e ^ {2 \pi i/h } $;

4) the product of any eigen value of $ A $ by $ \omega $ is an eigen value of $ A $;

5) for $ h > 1 $ one can find a permutation of the rows and columns that reduces $ A $ to the form

$$ \left \|

where $ A _ {j} $ is a matrix of order $ nh ^ {-} 1 $.

O. Perron proved the assertions 1) and 2) for positive matrices in [1], while G. Frobenius [2] gave the full form of the theorem.

References

[1] O. Perron, "Zur Theorie der Matrizen" Math. Ann. , 64 (1907) pp. 248–263
[2] G. Frobenius, "Ueber Matrizen aus nicht negativen Elementen" Sitzungsber. Königl. Preuss. Akad. Wiss. (1912) pp. 456–477
[3] F.R. [F.R. Gantmakher] Gantmacher, "The theory of matrices" , 1 , Chelsea, reprint (1977) (Translated from Russian)

Comments

The Perron–Frobenius theorem has numerous applications, cf. [a1], [a2].

References

[a1] E. Seneta, "Nonnegative matrices" , Allen & Unwin (1973)
[a2] K. Lancaster, "Mathematical economics" , Macmillan (1968)
How to Cite This Entry:
Perron-Frobenius theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Perron-Frobenius_theorem&oldid=48164
This article was adapted from an original article by D.A. Suprunenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article