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. ${F_x}(n)$) for the maximum number of $n \times n$ matrices (resp. $n \times n$ 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 $n = (2a + 1){2^b}$, where $b = c + 4d$ and a, b, c, d are nonnegative integers with $0 \leqq c < 4$, and define the Hurwitz-Radon function $\rho$ of n as $\rho (n) = {2^c} + 8d$. It is known [l], [2] that \[ \begin {array}{*{20}{c}} {R(n) = \rho (n),C(n) = 2b + 2,H(n) = \rho \left ( {\frac {1}{2}n} \right ) + 4,} \\ {{F_h}(n) = {\text {F}}\left ( {\frac {1}{2}n} \right ) + 1,{\text {for}}\;F = R,C\;{\text {or}}\;H,} \\ \end {array} \] where $\rho (\tfrac {1}{2}n) = F(\tfrac {1}{2}n) = 0$ if n is odd. In this note we use these known results to prove the following theorems.