# Perron-Frobenius theorem

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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,

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%E2%80%93Frobenius_theorem&oldid=22894