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APRG Seminar

Title: Totally positive matrices
Speaker: T.E.S. Raghavan (University of Illinois at Chicago, USA)
Date: 18 February 2025
Time: 2:30 pm
Venue: LH-1, Mathematics Department

An $m \times n$ matrix $A = (a_{ij})$ is called a nonnegative matrix ($\geq 0$) when $a_{ij} \geq 0\ \forall i=1,\dots,m; j = 1,\dots,n$. The matrix $A$ is called totally nonnegative (TN) if all minors of all order are nonnegative. The matrix $A$ is called totally positive (TP) if all minors of order $q$, $q = 1, 2, \dots, \min(m, n)$ are strictly positive. When $m=n$, the matrix $A$ is called oscillatory if $A$ is TN and for some power $k$ $A^k$ is TP.

While the Russian school headed by M.G. Krein and F.R. Gantmacher were motivated by problems of small oscillations and vibrations of mechanical systems that led to the study of the spectral properties of oscillatory matrices, the American school was headed by I.J. Schoenberg and A.M. Whitney who concentrated on the sign variation properties of arbitrary vectors $y \in \mathbb{R}^n$ and their image $Ay$ under TN matrices. Influenced by the works of Krein and Schoenberg, Samuel Karlin noticed its applicability to the study of the properties of the transition function of strong Markov processes, and their intimate connections to the count of disjoint paths among paths in a directed graph without cycles. Yet another application appears in statistical decision theory. Virtually many standard one parameter family of probability density functions possess variation diminishing properties. Extensions of the Perron Frobenius theorem to Positive operators by M.G. Krein and M.A. Rutman, are also extendable to the study of totally positive operators.


Contact: +91 (80) 2293 2711, +91 (80) 2293 2265 ;     E-mail: chair.math[at]iisc[dot]ac[dot]in
Last updated: 17 Feb 2025