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A bound on the scrambling index of a primitive matrix using Boolean rank

✍ Scribed by Mahmud Akelbek; Sandra Fital; Jian Shen

Publisher: Elsevier Science
Year: 2009
Tongue: English
Weight: 153 KB
Volume: 431
Category: Article
ISSN: 0024-3795
DOI: 10.1016/j.laa.2009.06.031

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✦ Synopsis

The scrambling index of an n × n primitive matrix A is the smallest positive integer k such that A k (A t ) k = J, where A t denotes the transpose of A and J denotes the n × n all ones matrix. For an m × n Boolean matrix M, its Boolean rank b(M) is the smallest positive integer b such that M = AB for some m × b Boolean matrix A and b × n Boolean matrix B. In this paper, we give an upper bound on the scrambling index of an n × n primitive matrix M in terms of its Boolean rank b(M). Furthermore we characterize all primitive matrices that achieve the upper bound.

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