Exponents of indecomposability

Abstract

Let r, n be integers, − n < r < n. An n × n matrix A is called r-indecomposable if it contains no k × l zero submatrix with k + l = n − r + l. If A is primitive, then there is a smallest positive integer, h r ∗(A) , such that A m is r-indecomposable for all m ⩾ h r ∗ (A) . The integer h r ∗ (A) is called the strict exponent of r-indecomposability of the primitive matrix A. It refines the well-known exponent, exp (A) = h n−1 ∗ (A) . Brualdi and Liu (Czechoslovak Math. J. 40 115 (1990) 659–670; Proc. Amer. Math. Soc. 112 (4) (1991) 1193–1201) conjectured that h O ∗(A) ⩽ [n 2/4] and h l ∗(A) ⩽ [(n + 1) 2/4] . We show that h r ∗(A) ⩽ max “1, s(n − s + r − 1) + 1” where s is the smallest positive integer such that trace ( A s ) > 0. This improves the conjectured bounds for h O ∗ and h l ∗ .