Date of Award
5-9-2016
Degree Type
Dissertation
Degree Name
Doctor of Philosophy (PhD)
Department
Mathematics and Statistics
First Advisor
Frank Hall
Second Advisor
Marina Arav
Third Advisor
Zhongshan Li
Fourth Advisor
Hendricus van der Holst
Abstract
The first part of this dissertation answers the questions posed in the article ``A note on permanents and generalized complementary basic matrices", {\it Linear Algebra Appl.} 436 (2012), by M. Fiedler and F. Hall. Further results on permanent compounds of generalized complementary basic matrices are obtained. Most of the results are also valid for the determinant and the usual compound matrix. Determinant and permanent compound products which are intrinsic are also considered, along with extensions to total unimodularity.
The second part explores some connections of dense alternating sign matrices with total unimodularity, combined matrices, and generalized complementary basic matrices.
In the third part of the dissertation, an explicit formula for the ranks of dense alternating sign matrices is obtained. The minimum rank and the maximum rank of the sign pattern of a dense alternating sign matrix are determined. Some related results and examples are also provided.
DOI
https://doi.org/10.57709/8480656
Recommended Citation
Stroev, Mikhail, "Some Results on Generalized Complementary Basic Matrices and Dense Alternating Sign Matrices." Dissertation, Georgia State University, 2016.
doi: https://doi.org/10.57709/8480656