Inequalities for totally nonnegative matrices: Gantmacher-Krein, Karlin, and Laplace. (English) Zbl 07850925
This paper focuses on determinantal inequalities for totally non-negative (TN) matrices, which are real matrices with all minors being non-negative. The authors introduce index-row and index-column operations that apply to determinantal inequalities for TN matrices, resulting in new additive inequalities embedded within classical identities by Laplace and Karlin. Additionally, they explore multiplicative inequalities and provide a novel classification of these inequalities, enhancing the classification by S. M. Fallat et al. [Adv. Appl. Math. 30, No. 3, 442–470 (2003; Zbl 1030.15021)]
and M. Skandera [J. Algebr. Comb. 20, No. 2, 195–211 (2004; Zbl 1066.05089)].
The paper also presents an algorithm that identifies certain determinantal expressions that do not form an inequality for TN matrices and characterizes multiplicative inequalities comparing products of pairs of minors. The authors describes in details the connection between TN matrices and planar networks, using this relationship to prove their results. They conclude with a discussion on the Barrett-Johnson determinantal inequality and its implications for future research in the field. This paper is well-written, and provides interesting results in the area of determinantal inequalities and totally non-negative matrices.
The paper also presents an algorithm that identifies certain determinantal expressions that do not form an inequality for TN matrices and characterizes multiplicative inequalities comparing products of pairs of minors. The authors describes in details the connection between TN matrices and planar networks, using this relationship to prove their results. They conclude with a discussion on the Barrett-Johnson determinantal inequality and its implications for future research in the field. This paper is well-written, and provides interesting results in the area of determinantal inequalities and totally non-negative matrices.
Reviewer: Ludovick Bouthat (Québec)
MSC:
| 15A45 | Miscellaneous inequalities involving matrices |
| 15A15 | Determinants, permanents, traces, other special matrix functions |
| 15B48 | Positive matrices and their generalizations; cones of matrices |
| 39B62 | Functional inequalities, including subadditivity, convexity, etc. |
Keywords:
totally non-negative matrices; planar networks; inequalities; identities; preservers; set theoretic operations; index operationsReferences:
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