Exponents of indecomposability

✍ Scribed by Jian Shen; David Gregory; Stewart Neufeld

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 1019 KB
• Volume: 288
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/s0024-3795(98)10213-6

✦ Synopsis

Let r, n be integers, -n < r < n. An n x n matrix A is called r-indecomposable if it contains no k x I zero submatrix with k + ! = n -r + 1. If A is primitive, then there is a smallest positive integer, h~.(A), such that A'" is r-indecomposable for all m >1 hi'.(A). The integer h,: (A) is called the strict exponent of r-indecomposability of the primitive matrix A. It refines the well-kno~n exponent, exp(A) = h,; j(,4).

Bruaidi and Liu (Czechoslovak Math. J. 4C 115 (1990)659-670; Proc. Amer. Math. Soc. 112 (4) (1991) 1193-1201) conjectured that h;j(A)<,[n"/4j and h~(.4) ~< L(n + 1)"/4]. We show that h~.(A) <~ max{I, s(n-s + r-1) + 1} where s is the smallest positive integer such that trace(A') > 0. This improves the conjectured bounds for h~ and hi.