Theoretical Statistics and Mathematics Unit, ISI Delhi

Inertia of Loewner Matrices

by Rajendra Bhatia, Shmuel Friedland and Tanvi Jain

Given positive numbers $p_1 < p_2 < \cdots < p_n,$ and a real number $r$ let
$L_r$ be the $n \times n$ matrix with its $i,j$ entry equal to
$(p_i^r-p_j^r)/(p_i-p_j).$ A
well-known theorem of C. Loewner says that $L_r$ is positive definite when $0 <
r < 1.$ In contrast, R. Bhatia and J. Holbrook, (Indiana Univ. Math. J, 49
(2000) 1153-1173) showed that when $1 < r < 2,$ the matrix $L_r$ has only one
positive eigenvalue, and made a conjecture about the signatures of eigenvalues
of $L_r$ for other $r.$ That conjecture is proved in this paper.

isid/ms/2015/12 [fulltext]

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