Inertia of Loewner Matrices
Classical Analysis and ODEs
2015-01-08 v1
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.
Keywords
Cite
@article{arxiv.1501.01505,
title = {Inertia of Loewner Matrices},
author = {Rajendra Bhatia and Shmuel Friedland and Tanvi Jain},
journal= {arXiv preprint arXiv:1501.01505},
year = {2015}
}
Comments
11 pages, 1 figure