A probabilistic representation of totally positive matrices
type: Item
instance of: scholarly article
title: A probabilistic representation of totally positive matrices
zbMATH Open document ID: 0611.60058
DOI: 10.1016/0196-8858(86)90035-7
author: Gerald S. Goodman
published in: Advances in Applied Mathematics
publication date: 1986-01-01T00:00:00Z
review text: \textit{C. Loewner} [Math. Z. 63, 338-340 (1955; Zbl 0068.250)] has shown that every non-singular totally positive matrix of finite order can be reached from the identity by forward solutions of the control equation \(dR/dt=C(t)R\), when C(t) varies in the class of Jacobi matrices with non- negative off-diagonal entries. We give a probabilistic solution of this equation by associating with it a finite-state non-stationary birth-and- death process and using the Feynman-Kac formula. This yields a probabilistic representation of non-singular totally positive matrices in terms of random evolutions. We then extend the representation to compound matrices by making use of a multi-particle system due to \textit{R. P. Feynman} [Phys. Rev., II. Ser. 76, 749-759 (1949; Zbl 0037.124), ibid. 76, 769-789 (1949; Zbl 0038.133)] and \textit{S. Karlin} and \textit{J. McGregor} [Pac. J. Math. 9, 1141-1164 (1959; Zbl 0092.345)]. As applications, we obtain both the Jacobi-Liouville formula and the inequality of \textit{F. R. Gantmacher} and \textit{M. G. Krein} [Oszillationsmatrizen, Oszillationskerne und kleine Schwingungen mechanischer Systeme. Akademie-Verlag (1964)] for the principal minors of totally positive matrices by simple probabilistic arguments.
Mathematics Subject Classification ID: 60H99
Mathematics Subject Classification ID: 15B52
Mathematics Subject Classification ID: 93E20
zbMATH DE Number: 3988396
zbMATH Keywords: finite-state non-stationary birth-and-death process
zbMATH Keywords: Feynman-Kac formula
zbMATH Keywords: totally positive matrices
MaRDI profile type: Publication
full work available at URL: 0196-8858(86)90035-7
OpenAlex ID: W2026945146
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description: scientific article; zbMATH DE number 3988396