The Exponent Set of Primitive, Nearly Reducible Matrices

Abstract

In [1] and [2], R. A. Brualdi and J. A. Ross studied the exponent set of a particular class of primitive matrices—primitive, nearly reducible matrices. They obtained an upper bound on the exponent and constructed some matrices with small exponents. Ross [2] suggested considering the problem of determining the quantity $e(n)$—the least integer $e(n)\geqq 6$ such that no $n \times n$ primitive, nearly reducible matrix has exponent $e(n)$. In this paper we give a nontrivial lower bound $e ( n )\geqq ( n^2 - 2n + 10 ) / 9 + 1$ by showing that every integer k with $6\leqq k\leqq ( n^2 - 2n + 10 ) / 9$ is an exponent of some $n \times n$ primitive, nearly reducible matrix. This also extends the result ([2, § 3]) that every integer k with $6\leqq k\leqq n + 1$ is the exponent of some $n \times n$ primitive, nearly reducible matrix.