Positivity and conditional positivity of Loewner matrices
Using the following well-known facts (a) the Cauchy matrix \(C = \left[\frac{1}{p_i} + \frac{1}{p_j}\right]\) is psd (positive semidefinite) for any \(n\) positive numbers \(p_1 ,\dots,p_n\), and (b) the Hadamard (entrywise) product of two psd matrices is also psd, in this paper it is elementarily shown that the Loewner matrices \(L = \left[\frac{f(p_i) - f(p_j)}{p_i - p_j} \right]\) for \(f(t) = t^r\), \(t \in (0,\infty)\), are (1) psd, (2) conditionally negative definite (cnd), and (3) conditionally positive definite (cpd) for \(r\) in \([0,1]\), \([1,2]\), and \([2,3]\), respectively. Next, for \(f\) a real valued function on any open interval, the following assertions are examined: (i) \(L\) are psd if \(f\) is operator monotone, (ii) \(L\) are cnd if \(f\) is operator convex, and (iii) \(L\) are cpd if \(f(t) = tg(t)\) for an operator convex function \(g\). It is concluded that (i) and (iii) are true, and (ii) is not true.
scientific article; zbMATH DE number 5797571