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Difference between revisions of "Perron-Frobenius theorem"

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Let a real square  $  ( n \times n) $-
+
Let a real square  $  ( n \times n) $-matrix  $  A $
matrix  $  A $
 
 
be considered as an operator on  $  \mathbf R  ^ {n} $,  
 
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} $
+
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
 
be its eigen values, enumerated such that
  
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with positive coordinates corresponding to  $  r $;
 
with positive coordinates corresponding to  $  r $;
  
3) the numbers  $  \lambda _ {1} \dots \lambda _ {h} $
+
3) the numbers  $  \lambda _ {1}, \dots, \lambda _ {h} $
coincide, apart from their numbering, with the numbers  $  r, \omega r \dots \omega  ^ {h-} 1 r $,  
+
coincide, apart from their numbering, with the numbers  $  r, \omega r, \dots, \omega  ^ {h-1} r $,  
 
where  $  \omega = e ^ {2 \pi i/h } $;
 
where  $  \omega = e ^ {2 \pi i/h } $;
  
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\begin{array}{ccccc}
 
\begin{array}{ccccc}
  0  &A _ {1}  & 0  &\dots & 0  \\
+
  0  &A _ {1}  & 0  &\cdots & 0  \\
  0  & 0  &A _ {2}  &\dots & 0  \\
+
  0  & 0  &A _ {2}  &\cdots & 0  \\
\dots &\dots &\dots &\dots &\dots \\
+
\vdots &\vdots &\vdots &\ddots &\vdots \\
  0  & 0  & 0  &\dots &A _ {h-} 1 \\
+
  0  & 0  & 0  &\cdots &A _ {h-1}  \\
A _ {h}  & 0  & 0  &\dots & 0  \\
+
A _ {h}  & 0  & 0  &\cdots & 0  \\
 
\end{array}
 
\end{array}
 
  \right \| ,
 
  \right \| ,
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where  $  A _ {j} $
 
where  $  A _ {j} $
is a matrix of order  $  nh  ^ {-} 1 $.
+
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.

Latest revision as of 04:05, 4 March 2022


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

$$ \left \| \begin{array}{ccccc} 0 &A _ {1} & 0 &\cdots & 0 \\ 0 & 0 &A _ {2} &\cdots & 0 \\ \vdots &\vdots &\vdots &\ddots &\vdots \\ 0 & 0 & 0 &\cdots &A _ {h-1} \\ A _ {h} & 0 & 0 &\cdots & 0 \\ \end{array} \right \| , $$

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=49522
This article was adapted from an original article by D.A. Suprunenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article