The signed enhanced principal rank characteristic sequence

TL;DR

This paper introduces the signed enhanced principal rank characteristic sequence (sepr-sequence) for Hermitian matrices, providing a detailed classification and identifying unattainable sequences, with complete results for small matrices and certain real symmetric cases.

Contribution

It defines the sepr-sequence, extends the existing epr-sequence concept, and characterizes attainable sequences, including prohibiting certain subsequences and classifying small cases.

Findings

Sepr-sequences offer more detailed information than traditional epr-sequences.

Certain subsequences like A*N and NA* are proven unattainable for Hermitian matrices.

Complete classification of sepr-sequences for matrices of order 1 to 3.

Abstract

The signed enhanced principal rank characteristic sequence (sepr-sequence) of an $n \times n$ Hermitian matrix is the sequence $t_1t_2 \cdots t_n$, where $t_k$ is either $\tt A^*$, $\tt A^+$, $\tt A^-$, $\tt N$, $\tt S^*$, $\tt S^+$, or $\tt S^-$ based on the following criteria: $t_k = \tt A^*$ if $B$ has both a positive and a negative order-$k$ principal minor, and each order-$k$ principal minor is nonzero. $t_k = \tt A^+$ (respectively, $t_k = \tt A^-$) if each order-$k$ principal minor is positive (respectively, negative). $t_k = \tt N$ if each order-$k$ principal minor is zero. $t_k = \tt S^*$ if $B$ has each a positive, a negative, and a zero order-$k$ principal minor. $t_k = \tt S^+$ (respectively, $t_k = \tt S^-$) if $B$ has both a zero and a nonzero order-$k$ principal minor, and each nonzero order-$k$ principal minor is positive (respectively, negative). Such sequences provide more information than the $({\tt A,N,S})$ epr-sequence in the literature, where the $k$th term is either $\tt A$, $\tt N$, or $\tt S$ based on whether all, none, or some (but not all) of the order-$k$ principal minors of the matrix are nonzero. Various sepr-sequences are shown to be unattainable by Hermitian matrices. In particular, by applying Muir's law of extensible minors, it is shown that subsequences such as $\tt A^*N$ and $\tt NA^*$ are prohibited in the sepr-sequence of a Hermitian matrix. For Hermitian matrices of orders $n=1,2,3$, all attainable sepr-sequences are classified. For real symmetric matrices, a complete characterization of the attainable sepr-sequences whose underlying epr-sequence contains $\tt ANA$ as a non-terminal subsequence is established.