On a question of Bhatia, Friedland and Jain II.

Let $ p_1 \lt p_2 \lt \cdots \lt p_n $ 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} $ x r + 1 given by $ L_{r+1}=\begin {bmatrix}\frac {p_i^{r+1}-p_j^{r+1}}{p_i-p_j}\end {bmatrix} $ L r + 1 = [ p i r + 1 − p j r + 1 p i − p j ] and matrix $ P_r=[\begin {smallmatrix}{(p_i+p_j)^r}\end {smallmatrix}] $ 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} $ L r + 1 and $ P_r $ P r . Indeed, a non-singular matrix X over $ \mathbb {R} $ R is explicitly obtained such that $ X'P_rX=L_{r+1} $ X ′ P r X = L r + 1 in this paper. [ABSTRACT FROM AUTHOR]