## Mathematics Dissertations

5-9-2016

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics and Statistics

Frank Hall

Marina Arav

Zhongshan Li

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.

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