On Spaces of Matrices Containing a Nonzero Matrix of Bounded Rank
نویسندگان
چکیده
Let Mn(R) and Sn(R) be the spaces of n × n real matrices and real symmetric matrices respectively. We continue to study d(n, n − 2,R): the minimal number such that every -dimensional subspace of Sn(R) contains a nonzero matrix of rank n−2 or less. We show that d(4, 2,R) = 5 and obtain some upper bounds and monotonicity properties of d(n, n − 2,R). We give upper bounds for the dimensions of n − 1 subspaces (subspaces where every nonzero matrix has rank n − 1) of Mn(R) and Sn(R), which are sharp in many cases. We study the subspaces of Mn(R) and Sn(R) where each nonzero matrix has rank n or n − 1. For a fixed integer q > 1 we find an infinite sequence of n such that any (q+1 2 ) dimensional subspace of Sn(R) contains a nonzero matrix with an eigenvalue of multiplicity at least q.
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