Inertia of Loewner Matrices

TL;DR

This paper proves a conjecture about the eigenvalue signatures of Loewner matrices L_r for various real numbers r, extending known results on their positive definiteness and spectral properties.

Contribution

The paper establishes the eigenvalue signature conjecture for Loewner matrices L_r across different ranges of r, advancing understanding of their spectral behavior.

Findings

L_r is positive definite for 0<r<1

L_r has exactly one positive eigenvalue for 1<r<2

Conjecture on eigenvalue signatures for other r is proved

Abstract

Given positive numbers p_1 < p_2 < ... < p_n, and a real number r let L_r be the n by n matrix with its (i,j) entry equal to (p_i^r-p_j^r)/(p_i-p_j). A well-known theorem of C. Loewner says that L_r is positive definite when 0 < r < 1. In contrast, R. Bhatia and J. Holbrook, (Indiana Univ. Math. J, 49 (2000) 1153-1173) showed that when 1 < r < 2, the matrix L_r has only one positive eigenvalue, and made a conjecture about the signatures of eigenvalues of L_r for other r. That conjecture is proved in this paper.