2023/08/02 by Xavier Martínez-Rivera, Martínez-Rivera, Xavier, Kamonchanok Saejeam +1
Computer Science · Engineering · Mathematics · #Advanced Topics in Algebra #FOS: Mathematics #Matrix Theory and Algorithms #Rings and Algebras (math.RA) #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2308.01397
openalex publication_date 2023/08/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The signed enhanced principal rank characteristic sequence (sepr-sequence) of a given n × n Hermitian matrix B is the sequence t1t2 ⋯ tn, where tk is \tt A^*, \tt A+, \tt A-, \tt N, \tt S^*, \tt S+, or \tt S-, based on the following criteria: tk = \tt A^* if all the order-k principal minors of B are nonzero, and two of those minors are of opposite sign; tk = \tt A+ (respectively, tk = \tt A-) if all the order-k principal minors of B are positive (respectively, negative); tk = \tt N if all the order-k principal minors of B are zero; tk = \tt S^* if B has a positive, a negative, and a zero order-k principal minor; tk = \tt S+ (respectively, tk = \tt S-) if B has both a zero and a nonzero order-k principal minor, and all the nonzero order-k principal minors of B are positive (respectively, negative). A complete characterization of the sequences of order 2 and order 3 that do not occur as a subsequence of the sepr-sequence of any Hermitian matrix is presented (a sequence has order k if it has k terms). An analogous characterization for real symmetric matrices is presented as well.