Theoretical Statistics and Mathematics Unit, ISI Delhi

Positivity and Conditional Positivity of Loewner Matrices

by 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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