Principal rank characteristic sequences

Abstract

<p>The necessity to know certain information about</p> <p>the principal minors of a given/desired</p> <p>matrix is a situation that</p> <p>arises in several areas of mathematics.</p> <p>As a result, researchers associated two</p> <p>sequences with an $n \times n$</p> <p>symmetric,</p> <p>complex Hermitian, or</p> <p>skew-Hermitian matrix $B$.</p> <p>The first of these is the</p> <p>principal rank characteristic sequence</p> <p>(abbreviated pr-sequence).</p> <p>This sequence is defined as</p> <p>$r_0]r_1 \cdots r_n$,</p> <p>where, for $k \geq 1$,</p> <p>$r_k = 1$ if $B$ has a</p> <p>nonzero order-$k$ principal minor, and</p> <p>$r_k = 0$, otherwise;</p> <p>$r_0 = 1$ if and only if</p> <p>$B$ has a $0$ diagonal entry.</p> <p>The second sequence, one that</p> <p>``enhances'' the pr-sequence, is the</p> <p>enhanced principal rank characteristic sequence (epr-sequence), denoted by</p> <p>$\ell_1 \ell_2 \cdots \ell_n$, where $\ell_k$ is either</p> <p>$\tt A$, $\tt S$, or $\tt N$, based on whether</p> <p>all, some but not all, or none of the</p> <p>order-$k$ principal minors of $B$ are nonzero.</p> <p>In this dissertation,</p> <p>restrictions for the attainability of</p> <p>epr-sequences by real symmetric matrices are established.</p> <p>These restrictions are then used to classify two related</p> <p>families of sequences that are attainable by real symmetric matrices:</p> <p>the family of pr-sequences</p> <p>not containing three consecutive $1$s, and</p> <p>the family of epr-sequences</p> <p>containing an $\tt{N}$ in every subsequence of</p> <p>length $3$.</p> <p>The epr-sequences that are attainable by symmetric matrices over fields of characteristic $2$ are considered:</p> <p>For the prime field of order $2$, a complete characterization of these epr-sequences is obtained;</p> <p>and for more general fields of characteristic $2$, some restrictions are also obtained.</p> <p>A sequence that refines the epr-sequence of</p> <p>a Hermitian matrix $B$, the</p> <p>signed enhanced principal rank characteristic sequence (sepr-sequence), is introduced.</p> <p>This sequence is defined as</p> <p>$t_1t_2 \cdots t_n$, where</p> <p>$t_k$ is either $\tt A^*$, $\tt A^+$, $\tt A^-$, $\tt N$, $\tt S^*$, $\tt S^+$, or $\tt S^-$, based on the following criteria:</p> <p>%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%</p> <p>$t_k = \tt A^*$ if $B$ has both a positive and a negative order-$k$ principal minor, and each order-$k$ principal minor is nonzero;</p> <p>%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%</p> <p>$t_k = \tt A^+$ (respectively, $t_k = \tt A^-$) if each order-$k$ principal minor is positive (respectively, negative);</p> <p>%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%</p> <p>$t_k = \tt N$ if each order-$k$ principal minor is zero;</p> <p>%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%</p> <p>$t_k = \tt S^*$ if $B$ has each a positive, a negative, and a zero order-$k$ principal minor;</p> <p>%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%</p> <p>$t_k = \tt S^+$ (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).</p> <p>The unattainability of various</p> <p>sepr-sequences is established.</p> <p>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.</p> <p>The notion of a nonnegative and nonpositive subsequence is introduced, leading to a connection with positive semidefinite matrices.</p> <p>Moreover, restrictions for sepr-sequences attainable by real symmetric matrices are established.</p>