Positivity and conditional positivity of Loewner matrices

Bhatia, Rajendra ; Sano, Takashi (2009) Positivity and conditional positivity of Loewner matrices Positivity . ISSN 1385-1292

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Official URL: http://www.springerlink.com/index/X1316J980240W785...

Related URL: http://dx.doi.org/10.1007/s11117-009-0027-2


We give elementary proofs of the fact that the Loewner matrices [f(pi)-f(pj)/pi-pj]corresponding to the function f(t) = t r on (0, ∞ ) 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, ∞ ) the Loewner matrices corresponding to an operator convex function on (-1, 1) need not be conditionally negative definite.

Item Type:Article
Source:Copyright of this article belongs to Springer-Verlag.
Keywords:Loewner Matrix; Operator Monotone; Operator Convex; Positive Semidefinite; Conditionally Positive Definite; Conditionally Negative Definite
ID Code:2528
Deposited On:08 Oct 2010 07:01
Last Modified:08 Oct 2010 07:01

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