Bounds on the k-th generalized base of a primitive sign pattern matrix |
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Authors: | Qian Li Jeffrey Stuart |
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Affiliation: | 1. Department of Mathematics , South China Agricultural University , Guangzhou, China;2. School of Mathematical Sciences , South China Normal University , Guangzhou, China;3. Department of Mathematics , Pacific Lutheran University , Tacoma, WA, USA |
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Abstract: | You et al. [L. You, J. Shao, and H. Shan, Bounds on the bases of irreducible generalized sign pattern matrices, Lin. Alg. Appl. 427 (2007), pp. 285–300] extended the concept of the base of a powerful sign pattern matrix to the nonpowerful, irreducible sign pattern matrices. The key to their generalization was to view the relationship A l =A l?+?p as an equality of generalized sign patterns rather than of sign patterns. You, Shao and Shan showed that for primitive generalized sign patterns, the base is the smallest positive integer k such that all entries of A k are ambiguous. In this paper we study the k-th generalized base for nonpowerful primitive sign pattern matrices. For a primitive, nonpowerful sign pattern A, this is the smallest positive integer h such that Ak has h rows consisting entirely of ambiguous entries. Extending the work of You, Shao and Shan, we obtain sharp upper bounds on the k-th generalized base, together with a complete characterization of the equality cases for those bounds. We also show that there exist gaps in the k-th generalized base set of the classes of such matrices. |
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Keywords: | nonpowerful sign pattern k-th generalized base generalized sign pattern powers of generalized sign patterns primitive sign pattern |
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