Hogben, Leslie; Lin, Jephian C.-H.; Olesky, D. D.; van den Driessche, P. The sepr-sets of sign patterns. (English) Zbl 1452.15018 Linear Multilinear Algebra 68, No. 10, 2044-2068 (2020). Summary: Given a real symmetric \(n\times n\) matrix, the sepr-sequence \(t_1\cdots t_n\) records information about the existence of principal minors of each order that are positive, negative, or zero. This paper extends the notion of the sepr-sequence to matrices whose entries are of prescribed signs, that is, to sign patterns. A sufficient condition is given for a sign pattern to have a unique sepr-sequence, and it is conjectured to be necessary. The sepr-sequences of sign semi-stable patterns are shown to be well-structured; in some special circumstances, the sepr-sequence is enough to guarantee that the sign pattern is sign semi-stable. In alignment with previous work on symmetric matrices, the sepr-sequences for sign patterns realized by symmetric nonnegative matrices of orders two and three are characterized. Cited in 1 ReviewCited in 1 Document MSC: 15B35 Sign pattern matrices 15A15 Determinants, permanents, traces, other special matrix functions 15B48 Positive matrices and their generalizations; cones of matrices 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.) Keywords:principal rank characteristic sequence; sign pattern; sign semi-stable pattern; principal minor; digraph PDFBibTeX XMLCite \textit{L. Hogben} et al., Linear Multilinear Algebra 68, No. 10, 2044--2068 (2020; Zbl 1452.15018) Full Text: DOI arXiv References: [1] Friedland, S., On inverse multiplicative eigenvalue problems for matrices, Linear Algebra Appl, 12, 127-137 (1975) · Zbl 0329.15003 [2] Griffin, K.; Tsatsomeros, M., Principal minors, Part I: a method for computing all the principal minors of a matrix, Linear Algebra Appl, 419, 107-124 (2006) · Zbl 1110.65034 [3] Holtz, O.; Schneider, H., Open problems on GKK τ-matrices, Linear Algebra Appl, 345, 263-267 (2002) · Zbl 1160.15305 [4] Oeding, L., Set-theoretic defining equations of the variety of principal minors of symmetric matrices, Algebr Number Theory, 5, 75-109 (2011) · Zbl 1238.14035 [5] Brualdi, RA; Deaett, L.; Olesky, DD, The principal rank characteristic sequence of a real symmetric matrix, Linear Algebra Appl, 436, 2137-2155 (2012) · Zbl 1236.15015 [6] Butler, S.; Catral, M.; Fallat, SM, The enhanced principal rank characteristic sequence, Linear Algebra Appl, 498, 181-200 (2016) · Zbl 1334.15017 [7] Martínez-Rivera, X., The signed enhanced principal rank characteristic sequence, Linear Multilinear Algebra, 66, 1484-1503 (2017) · Zbl 1392.15051 [8] Brualdi, RA; Shader, BL., Matrices of sign-solvable linear systems (1995), Cambridge: Cambridge University Press, Cambridge [9] Diestel, R., Graph theory (2017), Berlin: Springer, Berlin · Zbl 1375.05002 [10] Hall, FJ, Li, Z.Sign Pattern Matrices. In: Hogben, L., editor. Handbook of linear algebra, 2nd ed. Chapter 42. Boca Raton: CRC Press; 2014. p. 1-32. [11] Borobia, A.Inverse eigenvalue problems. In: Hogben, L., editor. Handbook of linear algebra, 2nd ed. Boca Raton: CRC Press; 2014. [12] Šmigoc, H., The inverse eigenvalue problem for nonnegative matrices, Linear Algebra Appl, 393, 365-374 (2004) · Zbl 1075.15012 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.