On Matrices Whose Real Linear Combinations Are Nonsingular

Let A be either the real field R, or the complex field C, or the skew field Q of quaternions. Let Au A2, • ■ ■ , Ak be nXn matrices with entries from A. Consider a typical linear combination E"-iV^> with real coefficients Xy; we shall say that the set {A¡} "has the property P" if such a linear combination is nonsingular (invertible) except when all the coefficients X> are zero. We shall write A(ra) for the maximum number of such matrices which form a set with the property P. We shall write Ah(») for the maximum number of Hermitian matrices which form a set with the property P. (Here, if A = i?, 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(ra), Aff(ra). 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 A over R; we will say that W "has the property P" if every nonzero w in W is nonsingular (invertible). We now ask for the maximum possible dimension of such a space. In [l], the first named author has proved that i?(ra) equals the socalled Radon-Hurwitz function, defined below. In this note we determine i?z/(ra), C(ra), Cj/(ra), Qin) and <2ff(ra) by deriving inequalities between them and i?(ra). 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 {A,} with the property P may be motivated as follows. For such a set, the system of partial differential equations