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Principal rank characteristic sequences

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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.