Abstract
Let f be a positive, continuously differentiable function on (0, ∞). We consider matrices K
f
of the form We show that for such a function f with f(0) = f ′(0) = 0, all K
f
are conditionally negative definite if and only if all
are positive semidefinite (p.s.d.), and give the integral representation of f. We also study functions f with K
f
(·, ·) conditionally negative definite or p.s.d.
Acknowledgements
The authors thank Professor R. Bhatia for the information on Citation1. The second author is supported by Grant-in-Aid for Scientific Research (C) [KAKENHI] 20540152.