# Inertia of the matrix $[(p_{i}+p_{j})_{r}]$

### Rajendra Bhatia

Indian Statistical Institute, New Delhi, India### Tanvi Jain

Indian Statistical Institute, New Delhi, India

## Abstract

Let $p_{1},p_{2},…,p_{n}$ be positive real numbers. It is well known that for all $r<0$ the matrix $[(p_{i}+p_{j})_{r}]$ is positive definite. Our main theorem gives a count of the number of positive and negative eigenvalues of this matrix when $r>0$: Connections with some other matrices that arise in Loewner’s theory of operator monotone functions and in the theory of spline interpolation are discussed.

## Cite this article

Rajendra Bhatia, Tanvi Jain, Inertia of the matrix $[(p_{i}+p_{j})_{r}]$. J. Spectr. Theory 5 (2015), no. 1, pp. 71–87

DOI 10.4171/JST/91