Abstract

J. C. Holladay and Richard S. Varga, On powers of non-negative matrices, Proc. Amer. Math. Soc. 9 (1958), 631-634. MR 20, 3885. Zbl. 96, p. 8.

Let A=||a_i,j || be an n x n matrix consisting of non-negative elements. It is well known [1, p. 463] that A is primitive if and only if, for some positive integer n, Aⁿ has all its elements positive. One needs to know only this property of primitive matrices to understand this paper. If A^k is positive (i.e. has all its elements positive), then A^h is also positive for all  integers h>k [1, p. 463]. 2 Letting A be primitive, we shall define γ(A) as the smallest positive integer h such that A_h is positive.