Inertia of Kwong matrices

TL;DR

This paper investigates the eigenvalue signatures of Kwong matrices, a class of matrices defined by a parameter r and positive numbers, extending previous work on Loewner matrices to understand their inertia.

Contribution

The paper determines the eigenvalue signatures of Kwong matrices for all real r, generalizing known results for Loewner matrices and advancing understanding of matrix inertia.

Findings

Eigenvalue signatures depend on the parameter r and the positive numbers p_i.

The results extend the spectral analysis from Loewner matrices to Kwong matrices.

The signatures are explicitly characterized for all real r.

Abstract

Let $r$ be any real number and for any $n$ let $p_1,\ldots,p_n$ be distinct positive numbers. A Kwong matrix is the $n\times n$ matrix whose $(i,j)$ entry is $(p_i^r+p_j^r)/(p_i+p_j).$ We determine the signatures of eigenvalues of all such matrices. The corresponding problem for the family of Loewner matrices $\begin{bmatrix}(p_i^r-p_j^r)/(p_i-p_j)\end{bmatrix}$ has been solved earlier.