2008/06/16 by H. Tracy Hall, Hall, H. Tracy, Charles R. Johnson +1 · 1 citation
Computer Science · Decision Sciences · Physics and Astronomy · #05B35 #15A15 #15A45 #15A48 #Advanced Mathematical Theories and Applications #Combinatorics (math.CO) #FOS: Mathematics #Matrix Theory and Algorithms #Metric Geometry (math.MG) #Multi-Criteria Decision Making #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.0806.2645
openalex publication_date 2008/06/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Considered is the multiplicative semigroup of ratios of products of principal minors bounded over all positive definite matrices. A long history of literature identifies various elements of this semigroup, all of which lie in a sub-semigroup generated by Hadamard-Fischer inequalities. Via cone-theoretic techniques and the patterns of nullity among positive semidefinite matrices, a semigroup containing all bounded ratios is given. This allows the complete determination of the semigroup of bounded ratios for 4-by-4 positive definite matrices, whose 46 generators include ratios not implied by Hadamard-Fischer and ratios not bounded by 1. For n > 4 it is shown that the containment of semigroups is strict, but a generalization of nullity patterns, of which one example is given, is conjectured to provide a finite determination of all bounded ratios.