On eigenvalues of certain special matrices

Abstract

Let g : R → [ 0 , ∞ ) be a conditionally negative definite function and f : [ 0 , ∞ ) → [ 0 , ∞ ) be a Bernstein function. We prove that the function h = f ∘ g is conditionally negative definite and that for distinct real numbers p 1 , p 2 , … , p n the inertia of the matrix [ h ( p i − p j ) ] is ( 1 , 0 , n − 1 ) if f is non-linear and g ( x ) = 0 only for x = 0. We also present a new and simple proof to show that the matrix [ log ⁡ ( 1 − p i p j ) ] is negative definite for n distinct positive real numbers p i < 1 , ∀ i . These results supplement and unify earlier results proved by a number of authors including Dyn, Goodman and Michelli, Bhatia and Jain, Garg and Aujla, Garg and Agarwal for operator monotone functions.