Subspaces of symmetric matrices containing matrices with a multiple first eigenvalue

Abstract

Let \l be an (r -l)(2nr + 2)/2 dimensional subspace of rc x 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 CalabFs theorem. We extend these results to hermitian matrices.