The enhanced principal rank characteristic sequence over a field of characteristic 2

The enhanced principal rank characteristic sequence (epr-sequence) of an \(n \times n\) symmetric matrix over a field \(\mathbb{F}\) was recently defined as \(\ell_1 \ell_2 \cdots \ell_n\), where \(\ell_k\) is either \(\tt A\), \(\tt S\), or \(\tt N\) based on whether all, some (but not all), or non...

Full description

Saved in:
Bibliographic Details
Published in:arXiv.org 2017-09
Main Author: Martínez-Rivera, Xavier
Format: Article
Language:English
Subjects:
Integers
Mathematical analysis
Matrix methods
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:The enhanced principal rank characteristic sequence (epr-sequence) of an \(n \times n\) symmetric matrix over a field \(\mathbb{F}\) was recently defined as \(\ell_1 \ell_2 \cdots \ell_n\), 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 the matrix are nonzero. Here, a complete characterization of the epr-sequences that are attainable by symmetric matrices over the field \(\mathbb{Z}_2\), the integers modulo \(2\), is established. Contrary to the attainable epr-sequences over a field of characteristic \(0\), our characterization reveals that the attainable epr-sequences over \(\mathbb{Z}_2\) possess very special structures. For more general fields of characteristic \(2\), some restrictions on attainable epr-sequences are obtained.
ISSN:2331-8422
DOI:10.48550/arxiv.1608.07871