On matrices whose nontrivial real linear combinations are nonsingular.

Abstract

Let F be the real field R, the complex field C, or the skew field H of quaternions, and d{F) the real dimension of F. We shall write F(n) (resp. Fx(n)) for the maximum number of X matrices (resp. X matrices with property x) with elements in F whose nontrivial linear combinations with real coefficients are nonsingular and x will stand for hermitian (h), skew-hermitian (sk-h), symmetric (s), or skew-symmetric (sk-s). If is a positive integer, we write n = (2a4-l)2!l, where b =c+4d and a, b, c, d are nonnegative integers with 0gc<4, and define the Hurwitz-Radon function