Perron-Frobenius theorem

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)