Structured matrices, continued fractions, and root localization of polynomials

Abstract

We give a detailed account of various connections between several classes of objects: Hankel, Hurwitz, Toeplitz, Vandermonde and other structured matrices, Stietjes and Jacobi-type continued fractions, Cauchy indices, moment problems, total positivity, and root localization of univariate polynomials. Along with a survey of many classical facts, we provide a number of new results.

79 pages; new material added to the Introduction

References in corpus (3)

• The repulsive lattice gas, the independent-set polynomial, and the Lovász local lemma
• Hermite-Biehler, Routh-Hurwitz, and total positivity
• The inverse eigenvalue problem for symmetric anti-bidiagonal matrices

Cited by in corpus (13)

• An isospectral problem for global conservative multi-peakon solutions of the Camassa-Holm equation
• Total positivity of sums, Hadamard products and Hadamard powers: Results and counterexamples
• Total nonnegativity of infinite Hurwitz matrices of entire and meromorphic functions
• Statistics of work performed by optical tweezers with general time-variation of their stiffness
• Total nonnegativity of finite Hurwitz matrices and root location of polynomials
• The Hurwitz-type theorem for the regular Coulomb wave function via Hankel determinants
• Generalized Hurwitz matrices, generalized Euclidean algorithm, and forbidden sectors of the complex plane
• Hurwitz matrices of doubly infinite series
• Continued fraction expansions of Herglotz-Nevanlinna functions and generalized indefinite strings of Stieltjes type
• Spectral identities for Schrödinger operators
• Self-interlacing polynomials
• Direct and inverse spectral problems for a class of non-selfadjoint band matrices
• Hermite-Poulain theorems for linear finite difference operators