Degree Type

Dissertation

Date of Award

2017

Degree Name

Doctor of Philosophy

Department

Mathematics

Major

Applied Mathematics

First Advisor

Leslie Hogben

Abstract

The necessity to know certain information about

the principal minors of a given/desired

matrix is a situation that

arises in several areas of mathematics.

As a result, researchers associated two

sequences with an $n \times n$

symmetric,

complex Hermitian, or

skew-Hermitian matrix $B$.

The first of these is the

principal rank characteristic sequence

(abbreviated pr-sequence).

This sequence is defined as

$r_0]r_1 \cdots r_n$,

where, for $k \geq 1$,

$r_k = 1$ if $B$ has a

nonzero order-$k$ principal minor, and

$r_k = 0$, otherwise;

$r_0 = 1$ if and only if

$B$ has a $0$ diagonal entry.

The second sequence, one that

``enhances'' the pr-sequence, is the

enhanced principal rank characteristic sequence (epr-sequence), denoted by

$\ell_1 \ell_2 \cdots \ell_n$, where $\ell_k$ is either

$\tt A$, $\tt S$, or $\tt N$, based on whether

all, some but not all, or none of the

order-$k$ principal minors of $B$ are nonzero.

In this dissertation,

restrictions for the attainability of

epr-sequences by real symmetric matrices are established.

These restrictions are then used to classify two related

families of sequences that are attainable by real symmetric matrices:

the family of pr-sequences

not containing three consecutive $1$s, and

the family of epr-sequences

containing an $\tt{N}$ in every subsequence of

length $3$.

The epr-sequences that are attainable by symmetric matrices over fields of characteristic $2$ are considered:

For the prime field of order $2$, a complete characterization of these epr-sequences is obtained;

and for more general fields of characteristic $2$, some restrictions are also obtained.

A sequence that refines the epr-sequence of

a Hermitian matrix $B$, the

signed enhanced principal rank characteristic sequence (sepr-sequence), is introduced.

This sequence is defined as

$t_1t_2 \cdots t_n$, where

$t_k$ is either $\tt A^*$, $\tt A^+$, $\tt A^-$, $\tt N$, $\tt S^*$, $\tt S^+$, or $\tt S^-$, based on the following criteria:

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$t_k = \tt A^*$ if $B$ has both a positive and a negative order-$k$ principal minor, and each order-$k$ principal minor is nonzero;

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$t_k = \tt A^+$ (respectively, $t_k = \tt A^-$) if each order-$k$ principal minor is positive (respectively, negative);

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$t_k = \tt N$ if each order-$k$ principal minor is zero;

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$t_k = \tt S^*$ if $B$ has each a positive, a negative, and a zero order-$k$ principal minor;

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$t_k = \tt S^+$ (respectively, $t_k = \tt S^-$) if $B$ has both a zero and a nonzero order-$k$ principal minor, and each nonzero order-$k$ principal minor is positive (respectively, negative).

The unattainability of various

sepr-sequences is established.

Among other results, it is shown that subsequences such as $\tt A^*N$ and $\tt NA^*$ cannot occur in the sepr-sequence of a Hermitian matrix.

The notion of a nonnegative and nonpositive subsequence is introduced, leading to a connection with positive semidefinite matrices.

Moreover, restrictions for sepr-sequences attainable by real symmetric matrices are established.

Copyright Owner

Xavier Martinez-Rivera

Language

en

File Format

application/pdf

File Size

123 pages

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