# Positivity and conditional positivity of Loewner matrices

Positivity and conditional positivity of Loewner matrices We give elementary proofs of the fact that the Loewner matrices $${[\frac{f(p_i) - f (p_j)}{p_i-p_j}]}$$ 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Positivity and conditional positivity of Loewner matrices

, Volume 14 (3) – Jul 22, 2009
10 pages

/lp/springer_journal/positivity-and-conditional-positivity-of-loewner-matrices-g4IdlHHSJN
Publisher
Springer Journals
Subject
Mathematics; Econometrics; Calculus of Variations and Optimal Control; Optimization; Potential Theory; Operator Theory; Fourier Analysis
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-009-0027-2
Publisher site
See Article on Publisher Site

### Abstract

We give elementary proofs of the fact that the Loewner matrices $${[\frac{f(p_i) - f (p_j)}{p_i-p_j}]}$$ 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.

### Journal

PositivitySpringer Journals

Published: Jul 22, 2009

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