A note on minimum rank and maximum nullity of sign patterns

Date
2011-03-01
Authors
Hogben, Leslie
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Mathematics
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Abstract

The miminum rank of a sign pattern matrix is defined to be the smallest possible rank over all real matrices having the given sign pattern. The maximum nullity of a sign pattern is the largest possible nullity over the same set of matrices, and is equal to the number of columns minus the minimum rank of the sign pattern. Definitions of various graph parameters that have been used to bound maximum nullity of a zero-nonzero pattern, including path cover number and edit distacnce, are extended to sign patterns, and the SNS number is introduced to usefully generalized the triangle number to sign patterns. It is shown number, and maximum nullity, path cover number and edit distance are equal, providing a method to compute minimum rank for tree sign patterns. The minimum rank of small sign patterns is determined.

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This article is published as Hogben, Leslie. "A note on minimum rank and maximum nullity of sign patterns." The Electronic Journal of Linear Algebra 22 (2011): 203-213. DOI: 10.13001/1081-3810.1435. Posted with permission.

Keywords
Minimum rank, Maximum nullity, Sign pattern, Tree sign pattern, Asymmetric minimum rank, Path cover number, Edit distance, SNS sign pattern, SNS number, Ditree, Matrix
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