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

Abstract

Abstract. The enhanced principal rank characteristic sequence (epr-sequence) of an n × n symmetric matrix over a field F was recently defined as 1 2 · · · n, where k is either A, S, or 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 Z 2 , the integers modulo 2, is established. Contrary to the attainable epr-sequences over a field of characteristic 0, this characterization reveals that the attainable epr-sequences over Z 2 possess very special structures. For more general fields of characteristic 2, some restrictions on attainable epr-sequences are obtained.

The signed enhanced principal rank characteristic sequence

“…In the interest of brevity, the notation B I for det(B[I]) in [2] and [8] is adopted here (when I = ∅, B ∅ is defined to have the value 1). Moreover, when I = {i 1 , i 2 , .…”

“…There has been substantial work done on pr-and epr-sequences (see [2,3,4,5,6,7,8], for example). Here, we introduce a sequence that extends the pr-and epr-sequence, which we think remains tractable, while providing further help for working on the principal minor assignment problem for Hermitian matrices: Definition 1.1.…”

On the almost-principal minors of a symmetric matrix

“…(ii) Neither NA nor NS is a subsequence of q 1 q 2 • • • q n . Theorem 1.2 establishes a contrast between the epr-sequences and qpr-sequences of symmetric matrices, since we do not have a complete characterization such as the one in Theorem 1.2 for epr-sequences when the field F is not the prime field of order 2 (see [13]). The absence of such a characterization for epr-sequences is due to the difficulty in understanding epr-sequences containing NA or NS as subsequences.…”

“…A if all of the principal minors of order k are nonzero; S if some but not all of the principal minors of order k are nonzero; N if none of the principal minors of order k are nonzero (i.e., all are zero), where a minor of order k is the determinant of a k × k submatrix of B. After subsequent work on epr-sequences (see [6,8,12,13]), another sequence, one that refines the epr-sequence, called the signed enhanced principal rank characteristic sequence (sepr-sequence), was introduced by Martínez-Rivera in [14]. Recently, Fallat and Martínez-Rivera [7] extended the definition of the epr-sequence by also taking into consideration the almost-principal minors of the matrix, leading them to a new sequence, which we will define after introducing some terminology: For B ∈ F n×n and α, β ⊆ {1, 2, .…”

Combinatorial properties of the enhanced principal rank characteristic sequence over finite fields

“…Results concerning symmetric matrices over various fields, including constructions of matrices attaining certain epr-sequences, as well as results stating that certain subsequences cannot occur in the epr-sequence of a symmetric matrix, were presented in [4]. In [8], Martínez-Rivera established the following (complete) characterization of the epr-sequences of symmetric matrices over the finite field F = F 2 , where, for a given sequence…”