On matrices whose real linear combinations are non-singular

Abstract

Let A be either the real field R, or the complex field C, or the skew field Q of quaternions. Let A1, A2, , Ak be nXn matrices with entries from A. Consider a typical linear combination J= X1A with real coefficients Xi; we shall say that the set { A } the property if such a linear combination is nonsingular (invertible) except when all the coefficients Xi are zero. We shall write A(n) for the maximum number of such matrices which form a set with the property P. We shall write AH(n) for the maximum number of matrices which form a set with the property P. (Here, if A = R, the word Hermitian merely means symmetric; if A= Q it is defined using the usual conjugation in Q.) Our aim is to determine the numbers A(n), AH(n). Of course, it is possible to word the problem more invariantly. Let W be a set of matrices which is a vector space of dimension k over R; we will say that W the property if every nonzero w in W is nonsingular (invertible). We now ask for the maximum possible dimension of such a space. In [1], the first named author has proved that R(n) equals the socalled Radon-Hurwitz function, defined below. In this note we determine RH(n), C(n), CH(n), Q(n) and QH(n) by deriving inequalities between them and R(n). The elementary constructions needed to prove these inequalities can also be used to give a simplified description of the Radon-Hurwitz matrices. The study of sets of real symmetric matrices {Aj} with the property P may be motivated as follows. For such a set, the system of partial differential equations