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arXiv · 1608.07871

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

Xavier Martínez-Rivera
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Abstract

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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Xavier Martínez-Rivera. 2017-09-02. The enhanced principal rank characteristic sequence over a field of characteristic 2. https://doi.org/10.13001/1081-3810.3389

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