Subspaces of symmetric matrices containing matrices with a multiple first eigenvalue

Abstract

Let u be an (r - l)(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. © 1976 Pacific Journal of Mathematics. All rights reserved.