The quasi principal rank characteristic sequence

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 $\mathtt{A}$, $\mathtt{S}$, or $\mathtt{N}$, 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.

14 pages

References in corpus (4)

• The Geometry of Gaussoids
• Classification of families of pr- and epr-sequences
• The enhanced principal rank characteristic sequence over a field of characteristic 2
• The signed enhanced principal rank characteristic sequence