Proof of a conjecture about the exponent of primitive matrices Original Research Article

An n × n nonnegative matrix A is called primitive if for some positive integer k, every entry in the matrix Ak is positive (Ak much greater-than 0). The exponent of primitivity of A is defined to be γ(A) = min{k set membership, variant Z+ : Ak much greater-than 0}, where Z+ denotes the set of positive integers. The upper bound on γ(A) due to Wielandt is γ(A) ≤ (n − 1)2 + 1, and a better bound for γ(A) due to Hartwig and Neumann is γ(A) ≤ m(m − 1), where m is the degree of the minimal polynomial of A. Also, Hartwig and Neumann conjecture that γ(A) ≤ (m − 1)2 + 1, which had been suggested in 1984. In this paper, we prove this conjecture.