Inertia of the matrix $[(p_i+p_j{)}^r]$

Abstract

Let $p_1,p_2,\dots ,p_n$ 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.