Department Colloquium

Fri., Mar. 3, 2023 3:30 p.m.

Location: RIC 208

Speaker: Prateek Vishwakarma, University of Regina

Title: Identities and inequalities for totally nonnegative matrices: Gantmacher-Krein, Karlin, and Laplace (85 kB)

Abstract:

Matrix row operations preserving total nonnegativity translate into preserving determinantal inequalities for these matrices. In this talk, we introduce row operations that act directly on the determinantal inequalities and identities for totally nonnegative matrices, and yield expressions that potentially are novel inequalities and identities for these matrices.

These row operations and the elementary bidiagonal factorization of totally nonnegative matrices assist us in generalizing the inequalities by Gantmacher-Krein (1941) for these matrices. This further unravels a parallel essence regarding the matrix identities due to Karlin (1968) and due to Laplace (1772) for totally nonnegative matrices.

These refinements reveal an analogous underlying pattern: a sequence of inequalities oscillating about zero. This intrigued us to investigate if more such inequalities exist for totally nonnegative matrices. Using the identification of these matrices with planar networks and the inequality-preserving row operations introduced above, we obtain a novel class of these oscillating inequalities.