Let p(1),..., p(n) be positive real numbers. It is well known that for every 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.
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