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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On matrices whose nontrivial real linear combinations are nonsingular.


Author: Yik-hoi Au-yeung
Journal: Proc. Amer. Math. Soc. 29 (1971), 17-22
MSC: Primary 15.40
DOI: https://doi.org/10.1090/S0002-9939-1971-0274478-2
MathSciNet review: 0274478
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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{displaymath}\begin{array}{*{20}{c}} {R(n) = \rho (n),C(n) = 2b + 2,H(n) =... ...ht) + 1,{\text{for}}\;F = R,C\;{\text{or}}\;H,} \\ \end{array} \end{displaymath}

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.

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DOI: https://doi.org/10.1090/S0002-9939-1971-0274478-2
Keywords: Hurwitz-Radon function, maximum independent set of matrices over the real field
Article copyright: © Copyright 1971 American Mathematical Society