Bounded Ratios of Products of Principal Minors of Positive Definite Matrices

Details Bounded Ratios of Products of Principal Minors of Positive Definite Matrices Bounded Ratios of Products of Principal Minors of Positive Definite Matrices

2008-06-16

arxiv.org/abs/0806.2645v1

Mathematics

Combinatorics

7 pages

Scientific paper

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.

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