Subspaces of symmetric matrices containing matrices with a multiple first eigenvalue

Shmuel Friedland and Raphael Loewy

Vol. 62 (1976), No. 2, 389–399

DOI: 10.2140/pjm.1976.62.389

Abstract

Let 𝒰 be an (r − 1)(2n−r + 2)∕2 dimensional subspace of n × n real valued symmetric matrices. Then 𝒰 contains a nonzero matrix whose greatest eigenvalue is at least of multiplicity r, if 2 ≦ r ≦ n − 1. This bound is best possible. We apply this result to prove the Bohnenblust generalization of Calabi’s theorem. We extend these results to hermitian matrices.

Mathematical Subject Classification 2000

Primary: 15A57

Milestones

Received: 4 September 1975

Published: 1 February 1976