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} $ | ||
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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-} | + | 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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0 & 0 &A _ {2} &\dots & 0 \\ | 0 & 0 &A _ {2} &\dots & 0 \\ | ||
\dots &\dots &\dots &\dots &\dots \\ | \dots &\dots &\dots &\dots &\dots \\ | ||
| − | 0 & 0 & 0 &\dots &A _ {h-} | + | 0 & 0 & 0 &\dots &A _ {h-1} \\ |
A _ {h} & 0 & 0 &\dots & 0 \\ | A _ {h} & 0 & 0 &\dots & 0 \\ | ||
\end{array} | \end{array} | ||
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where $ A _ {j} $ | where $ A _ {j} $ | ||
| − | is a matrix of order $ nh ^ {-} | + | 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. | ||
Revision as of 04:04, 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
$$ \left \| \begin{array}{ccccc} 0 &A _ {1} & 0 &\dots & 0 \\ 0 & 0 &A _ {2} &\dots & 0 \\ \dots &\dots &\dots &\dots &\dots \\ 0 & 0 & 0 &\dots &A _ {h-1} \\ A _ {h} & 0 & 0 &\dots & 0 \\ \end{array} \right \| , $$
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) |
Perron-Frobenius theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Perron-Frobenius_theorem&oldid=49522