Two Results In Matrix Theory

Zhan has determined the maximum and minimum numbers of positive entries of imprimitive irreducible nonnegative matrices with a given imprimitivity index. In this paper, we prove that if a positive integer is between the maximum and minimum numbers, then there exists an irreducible nonnegative matrix A with the given imprimitivity index and with the given number of positive entries.In the other part of this paper, we prove an inequality for matrix, whose special case plays a key role in proving a conjecture of Zhan.