Bounded Ratios of Products of Principal Minors of Positive Definite Matrices

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dc.creatorHall, H. Tracy
dc.creatorJohnson, Charles R.
dc.date2008-06-16
dc.date.accessioned2026-07-07T09:44:51Z
dc.date.available2026-07-07T09:44:51Z
dc.descriptionConsidered 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.
dc.description7 pages
dc.identifierhttps://arxiv.org/abs/0806.2645
dc.identifierhttp://arxiv.org/abs/0806.2645
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/163018
dc.subjectCombinatorics
dc.subjectMetric Geometry
dc.subjectRings and Algebras
dc.subject15A45; 15A15, 15A48, 05B35
dc.titleBounded Ratios of Products of Principal Minors of Positive Definite Matrices
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