Principal rank characteristic sequences
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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.