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. Fz(n)) for the maximum number of nXn matrices (resp. n Xn 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 n is a positive integer, we write n = (2a+1)2b, where b =c+4d and a, b, c, d are nonnegative integers with 0 2 and H8k-8(1)= 0, Hsk -(2)= 4. COROLLARY. H8 (n) = H(n) for all n; H8k_,(n) = H(n) for all n > 2. From the above results, it may be interesting to note that Fh(n) and F8kh(n) can be expressed by the function F(m), while F8(n) and F.k8(n) can be expressed by p and d(F) except for two exceptional cases (H8k-5(1) and Hk-8(2)). The corollary follows immediately from Theorems 2 and 3, the expression of H(n), and the fact that p(8n) = p(4n) +8. We denote by M(n, F) (resp. M(n, F.)) the set of all n Xn matrices (resp. nXn matrices with property x) with elements in F. If X CM(n, F), we denote by XI and Xc the transpose and conjugate of X respectively. We shall use I e1, e2, e3} to denote the basis of H Received by the editors January 17, 1970. AMS 1970 subject classifications. Primary 15A30, 15A57; Secondary 15A33.