## Intervals of Special Sign Regular Matrices

2015
##### Series
Konstanzer Schriften in Mathematik; 341
##### Publication type
Working Paper/Technical Report
##### Abstract
We consider classes of n-by-n sign regular matrices, i.e., of matrices with the property that all their minors of fixed order k have one specified sign or are allowed also to vanish, k = 1, ... ,n. If the sign is nonpositive for all k, such a matrix is called totally nonpositive. The application of the Cauchon algorithm to nonsingular totally nonpositive matrices is investigated and a new determinantal test for these matrices is derived. Also matrix intervals with respect to the checkerboard partial ordering are considered. This order is obtained from the usual entry-wise ordering on the set of the n-by-n matrices by reversing the inequality sign for each entry in a checkerboard fashion. For some classes of sign regular matrices it is shown that if the two bound matrices of such a matrix interval are both in the same class then all matrices lying between these two bound matrices are in the same class, too.
510 Mathematics
##### Keywords
Sign regular matrix, totally nonnegative matrix, totally nonpositve matrix, Cauchon algorithm, checkerboard ordering, matrix interval
##### Cite This
ISO 690ADM, Mohammad, Jürgen GARLOFF, 2015. Intervals of Special Sign Regular Matrices. Available under: doi: 10.1080/03081087.2015.1090388
BibTex
@techreport{Adm2015Inter-31960,
year={2015},
doi={10.1080/03081087.2015.1090388},
series={Konstanzer Schriften in Mathematik},
title={Intervals of Special Sign Regular Matrices},
number={341},
note={Der Artikel wird erscheinen in: "Linear and Multilinear Algebra"}
}

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<dcterms:abstract xml:lang="eng">We consider classes of n-by-n sign regular matrices, i.e., of matrices with the property that all their minors of fixed order k have one specified sign or are allowed also to vanish, k = 1, ... ,n. If the sign is nonpositive for all k, such a matrix is called totally nonpositive. The application of the Cauchon algorithm to nonsingular totally nonpositive matrices is investigated and a new determinantal test for these matrices is derived. Also matrix intervals with respect to the checkerboard partial ordering are considered. This order is obtained from the usual entry-wise ordering on the set of the n-by-n matrices by reversing the inequality sign for each entry in a checkerboard fashion. For some classes of sign regular matrices it is shown that if the two bound matrices of such a matrix interval are both in the same class then all matrices lying between these two bound matrices are in the same class, too.</dcterms:abstract>
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