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.