Exponents of nonnegative matrix pairs (Q1870058)

The notions of primitivity and exponent of a square nonnegative matrix \(A\) are classical: \(A\) is primitive provided there is a nonnegative integer \(k\) such that \(A^k\) is entrywise positive and in the case \(A\) is primitive the exponent of \(A\) is the smallest such \(k\). \textit{E. Fornasini} and \textit{M. E. Valcher} [ibid. 263, 275-310 (1997; Zbl 0887.93033)] have extended the notion of primitivity to pairs \((A,B)\) of square nonnegative matrices of the same order. The pair \((A,B)\) is primitive provided there exist nonnegative integers \(h\) and \(k\) such that the sum of all products formed by words consisting of \(hA\)'s and \(kB\)'s is entrywise positive. The paper defines the exponent of a nonnegative matrix pair to be the smallest value of \(h+k\) over all such \(h\) and \(k\). It is then shown that the largest exponent of a primitive pair of \(n\) by \(n\) nonnegative matrices lies in the interval \([(n^3-5n^2)/2\), \((3n^3+2n^2 -2n)/2]\). In addition, the exponent of a pair of nonnegative matrices is related to properties of an associated two-dimensional dynamical system.