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...

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Published in:arXiv.org 2017-09
Main Author: Martínez-Rivera, Xavier
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Integers
Mathematical analysis
Matrix methods
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description 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.
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subjects Integers
Mathematical analysis
Matrix methods
title The enhanced principal rank characteristic sequence over a field of characteristic 2
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