Positivity and Conditional Positivity of Loewner Matrices
نویسندگان
چکیده
We give elementary proofs of the fact that the Loewner matrices [ f(pi)−f(pj) pi−pj ] corresponding to the function f(t) = t 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. 2000 Mathematics Subject Classification: 15A48, 47A63, 42A82.
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