Possible numbers of positive entries of imprimitive nonnegative matrices

Abstract

Zhan, X., Extremal numbers of positive entries of imprimitive nonnegative matrix, Linear Algebra Appl. (in press) has determined the maximum and minimum numbers of positive entries of imprimitive irreducible nonnegative matrices with a given imprimitivity index. Let σ( A ) denote the number of positive entries of a matrix A. Let M(n, k) and m(n, k) denote the maximum and minimum numbers of positive entries of imprimitive irreducible nonnegative matrices of order n with a given imprimitivity index k, respectively. In this article, we prove that for any positive integer d with m(n,k)≤ d ≤ M(n,k), there exists an n × n irreducible nonnegative matrix A with imprimitivity index k such that σ (A)=d.

Keywords:

• Irreducible nonnegative matrix
• Imprimitive matrix
• Possible numbers

Keywords:

• AMS Subject Classifications:
• 15A48
• 05D99

Acknowledgments

The research was supported by NSFC grant 10571060. We are grateful to Prof. Xingzhi Zhan and the referees for their kind help and valuable suggestions. The research was supported by NSFC grant 10571060.