Linear Algebra/Algebra Seminar: Martínez-Rivera

Speaker: Xavier Martínez-Rivera (Iowa State University)

Title: Principal rank characteristic sequences

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 talk, known results about pr- and epr-sequences are discussed. New restrictions for the attainability of epr-sequences by real symmetric matrices are presented. Particular attention will be paid to the epr-sequences that are attainable by symmetric matrices over fields of characteristic $2$: for the prime field of order $2$, a complete characterization of these epr-sequences is given.