  Let K be a (commutative) field with characteristic not 2, and V be a linear
subspace of n by n matrices that have at most two eigenvalues in K
(respectively, at most one non-zero eigenvalue in K). We prove that the
dimension of V is less than or equal to n(n-1)/2 provided that n be greater
than 2 (respectively, the dimension of V is less than or equal to n(n-1)/2).
  We also classify, up to similarity, the linear subspaces of n by n matrices
in which every matrix has at most two eigenvalues (respectively, at most one
non-zero eigenvalue) in an algebraic closure of K and which have the maximal
dimension among such spaces.
