On a question of Bhatia, Friedland and Jain II

Abstract

Let p 1 < p 2 < ⋯ < p n be positive numbers and r a non-negative real number. The Loewner matrix associated with the function x r + 1 given by L r + 1 = [ p i r + 1 − p j r + 1 p i − p j ] and matrix P r = [ ( p i + p j ) r ] (the Hadamard inverse of rth Hadamard power of well-known Cauchy matrix) have same inertia. A question was left open in Inertia of Loewner matrices. Indiana Univ Math J. 2016;65(4):1251–1261 by Bhatia, Friedland and Jain to find a connection between these two matrix families. We aim to answer this question firmly in terms of a congruence relation between L r + 1 and P r . Indeed, a non-singular matrix X over R is explicitly obtained such that X ′ P r X = L r + 1 in this paper.