COSAM » Events » 2017 » June » Linear Algebra/Algebra Seminar

 Linear Algebra/Algebra Seminar: Martínez-Rivera Time: Jun 27, 2017 (04:00 PM) Location: Parker Hall 228 Details: 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.

Last updated: 06/26/2017