Characterization of Determinantal Inequalities in Positivity Classes
by
Charles Johnson
College of William and Mary

Beginning with Hadamard's inequality for positive definite matrices in 1893, there has been considerable interest in inequalities among products of principal minors in a matrix of given type, for both pure and applied reasons. This has resulted in the discovery of particular families of inequalities for positive definite, totally positive, M-matrices, and inverse M-matrices, etc, by Fischer, Koteljanskii, Fan, Carlson, Marcus, Barrett and Johnson, etc over the last 100 years. Recently a program was begun to determine ALL such inequalities for each of given classes (and even all bounded ratios of products of principal minors). We report on the current progress in this program that has not only discovered new families of inequalities but characterized all inequalities in a number of cases. Combinatorial and cone theoretic ideas play an important role.

Date received: November 18, 1998

Copyright © 1998 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cabw-30.