Publications and Preprints

Positivity and Conditional Positivity of Loewner Matrices
Rajendra Bhatia and Takashi Sano
We give elementary proofs of the fact that the Loewner matrices $\left [\frac{f(p_i) - f (p_j)}{p_i-p_j} \right ]$ corresponding to the function $f(t) = t^r$ on $(0, \infty)$ are positive semidefinite, conditionally negative definite, and conditionally positive definite, for $r$ in $[0, 1], [1, 2],$ and $[2, 3],$ respectively. We show that in contrast to the interval $(0, \infty)$ the Loewner matrices corresponding to an operator convex function on $(-1, 1)$ need not be conditionally negative definite.

isid/ms/2009/09 [fulltext]

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