The quasi principal rank characteristic sequence

TL;DR

This paper introduces the quasi principal rank characteristic sequence (qpr-sequence) for symmetric matrices, extending principal rank sequences, and characterizes which sequences are attainable over fields of characteristic zero.

Contribution

It defines the qpr-sequence, establishes necessary conditions for its attainability, and provides a complete probabilistic characterization over characteristic zero fields.

Findings

Defined the qpr-sequence for symmetric matrices.

Established necessary conditions for qpr-sequence attainability.

Provided a probabilistic characterization of attainable sequences.

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.