Inertia_of_Loewner_Matrices.pdf (356.61 kB)

Inertia of Loewner matrices.

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journal contribution
posted on 08.12.2016 by Bhatia R, Friedland S, Jain T
Given positive numbers p_1 < p_2 < ... < p_n, and a real number r let L_r be the n by 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.

Funding

The work of R. Bhatia is supported by a J. C. Bose National Fellowship, of S. Friedland by the NSF grant DMS-1216393, and of T. Jain by a SERB Women Excellence Award.

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Publisher Statement

This is a copy of an article published in the Indiana University Mathematics Journal. © 2016 Indiana University Mathematics Journal Publications. https://arxiv.org/abs/1501.01505

Publisher

Indiana University Mathematics Journal

Language

en_US

issn

0022-2518

Issue date

01/01/2016

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