Inertia of Loewner matrices

Authors

Rajendra Bhatia, Indian Statistical Institute (Delhi Centre)
Shmuel Friedland, University of Illinois at Chicago
Tanvi Jain, Indian Statistical Institute (Delhi Centre)

Article Type

Research Article

Publication Title

Indiana University Mathematics Journal

Abstract

Given positive numbers p1 < p2 < ⋯ < pn and a real number r, let Lr be the n x n matrix with its i, j entry equal to (pir - pjr)/(pi - pj). A well-known theorem of C. Loewner says that Lr 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 Lr has only one positive eigenvalue, and made a conjecture about the signatures of eigenvalues of Lr for other values of r. That conjecture is proved in this paper.

First Page

1251

Last Page

1261

DOI

10.1512/iumj.2016.65.5869

Publication Date

1-1-2016

Comments

Open Access; Green Open Access

Recommended Citation

Bhatia, Rajendra; Friedland, Shmuel; and Jain, Tanvi, "Inertia of Loewner matrices" (2016). Journal Articles. 4260.
https://digitalcommons.isical.ac.in/journal-articles/4260