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The converse of the Cowling--Obrechkoff--Thron theorem

Devon N. Munger, Pietro Paparella

TL;DR

This work proves the converse of the Cowling--Obrechkoff--Thron theorem, addressing a gap in Kellogg's eigenvalue inequality for matrices with positive or nonnegative principal minors. By showing that for $\mu = r(\cos \alpha + i \sin \alpha)$ with $|\alpha| \ge \pi/n$ there exists a monic polynomial of degree $\le n$ with nonnegative (and, in key cases, positive) coefficients vanishing at $\mu$, the authors enable the reverse implication needed to derive Kellogg's spectrum conditions from polynomial constraints. The paper develops a suite of auxiliary results (notably the $Q_j$-polynomials) to construct polynomials with the required coefficient signs and demonstrates how to extend these constructions to degree $n$. The results solidify the theoretical foundation behind Kellogg's inequality and fill a gap in the literature on eigenvalue problems for $P$ and $P_0$ matrices.

Abstract

In this work, the converse of the Cowling--Obrechkoff--Thron theorem is established. In addition to its theoretical interest, the result fills a gap in the proof of Kellogg's celebrated eigenvalue inequality for matrices whose principal minors are positive or nonnegative.

The converse of the Cowling--Obrechkoff--Thron theorem

TL;DR

This work proves the converse of the Cowling--Obrechkoff--Thron theorem, addressing a gap in Kellogg's eigenvalue inequality for matrices with positive or nonnegative principal minors. By showing that for with there exists a monic polynomial of degree with nonnegative (and, in key cases, positive) coefficients vanishing at , the authors enable the reverse implication needed to derive Kellogg's spectrum conditions from polynomial constraints. The paper develops a suite of auxiliary results (notably the -polynomials) to construct polynomials with the required coefficient signs and demonstrates how to extend these constructions to degree . The results solidify the theoretical foundation behind Kellogg's inequality and fill a gap in the literature on eigenvalue problems for and matrices.

Abstract

In this work, the converse of the Cowling--Obrechkoff--Thron theorem is established. In addition to its theoretical interest, the result fills a gap in the proof of Kellogg's celebrated eigenvalue inequality for matrices whose principal minors are positive or nonnegative.
Paper Structure (4 sections, 9 theorems, 20 equations)

This paper contains 4 sections, 9 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Background & Motivation
  3. Ancillary Results
  4. Proof of the Main Result

Key Result

Theorem 1.1

If $\lambda = r(\cos\theta + i \sin\theta)\in \mathbb{C}$, with $\theta \in (0,2\pi]$, then $\lambda$ is an eigenvalue of a $P$ matrix if and only if If $\lambda \ne 0$, then $\lambda$ is an eigenvalue of an $n$-by-$n$$P_0$ matrix if and only if

Theorems & Definitions (15)

  • Theorem 1.1: Kellogg
  • Theorem 1.2: Cowling--Obrechkoff--Thron
  • Remark 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • Theorem 2.3
  • Lemma 3.1
  • proof
  • ...and 5 more