Exponents of indecomposability

AbstractLet 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, hr∗(A), such that Am is r-indecomposable for all m ⩾ hr∗ (A). The integer hr∗ (A) is called the strict exponent of r-indecomposability of the primitive matrix A. It refines the well-known exponent, exp (A) = hn−1∗ (A).Brualdi and Liu (Czechoslovak Math. J. 40 115 (1990) 659–670; Proc. Amer. Math. Soc. 112 (4) (1991) 1193–1201) conjectured that hO∗(A) ⩽ [n2/4] and hl∗(A) ⩽ [(n + 1)2/4]. We show that hr∗(A) ⩽ max “1, s(n − s + r − 1) + 1” where s is the smallest positive integer such that trace (As) > 0. This improves the conjectured bounds for hO∗ and hl∗