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arXiv · 1711.01003

The quasi principal rank characteristic sequence

Shaun M. Fallat·Xavier Martínez-Rivera
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Abstract

A minor of a matrix is quasi-principal if it is a principal or an almost-principal minor. The quasi principal rank characteristic sequence (qpr-sequence) of an $n\times n$ symmetric matrix is introduced, which is defined as $q_1 q_2 \cdots q_n$, where $q_k$ is $\tt A$, $\tt S$, or $\tt N$, according as all, some but not all, or none of its quasi-principal minors of order $k$ are nonzero. This sequence extends the principal rank characteristic sequences in the literature, which only depend on the principal minors of the matrix. A necessary condition for the attainability of a qpr-sequence is established. Using probabilistic techniques, a complete characterization of the qpr-sequences that are attainable by symmetric matrices over fields of characteristic $0$ is given.

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Shaun M. Fallat, Xavier Martínez-Rivera. 2017-11-03. The quasi principal rank characteristic sequence. https://doi.org/10.1016/j.laa.2018.02.024

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