Let a real square -matrix be considered as an operator on , 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 be its eigen values, enumerated such that Then,

1) the number is a simple positive root of the characteristic polynomial of ;

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

3) the numbers coincide, apart from their numbering, with the numbers , where ;

4) the product of any eigen value of by is an eigen value of ;

5) for one can find a permutation of the rows and columns that reduces to the form where is a matrix of order .

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

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

References