Inertia of Loewner Matrices
Abstract
Given positive numbers p1 < p2 < ... < pn, and a real number r let Lr be the n by 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 r. That conjecture is proved in this paper.
0
Turn this paper into a lesson
ArcXiv compiles a structured reading guide from this paper's metadata: plain-English importance, contributions, prerequisite concepts, which sections to read first, flashcards, and a quiz. Grounded in the abstract, never invented.