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A note on spaces of symmetric matrices. (English) Zbl 1126.15019

A vector subspace \(X\subseteq M_n\) is said to be of constant rank \(r\), provided that all its nonzero members have rank \(r\). A classical problem here is to determine how large such a subspace can be.
The present authors continue this kind of study. They confine themselves to real symmetric matrices, and obtain a sharp upper and lower bound on the maximal dimension of vector subspaces of constant rank \((n-1)\) and of constant rank \((n-2)\). Upper and lower bound differ only by \(1\). Moreover, conditions are supplied, which force optimality of lower, respectively, of upper bound.
Similar estimates are also obtained for complex Hermitian matrices.

MSC:

15A30 Algebraic systems of matrices
55N15 Topological \(K\)-theory
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