Shaun Fallat - Multiplicative principal minor inequalities for totally nonnegative matrices

An m-by-n matrix A is said to be totally nonnegative if the determinant of every square submatrix (i.e., minor) of A is nonnegative. Our main interest lay in characterizing all the inequalities that exist among products of principal minors (minors of A based on the same row and column index sets) of nonsingular totally nonnegative matrices. We give a complete characterization of all such inequalities for n-by-n totally nonnegative matrices with . We also illustrate certain key steps in the proof of this characterization, which offer interesting applications of symbolic computation, graph theory, convexity, and linear programming. Other results are presented including general conditions which guarantee when the product of two principal minors is less the product of two other principal minors, and many others.

This is joint work with Professors M. Gekhtman and C. Johnson.