Spaces of matrices with few eigenvalues

Let KK be a (commutative) field with characteristic not 2, and VV be a linear subspace of matrices of Mn(K)Mn(K) that have at most two eigenvalues in KK (respectively, at most one non-zero eigenvalue in KK). We prove that dimV⩽(n2)+2 provided that n⩾3n⩾3 (respectively, dimV⩽(n2)+1).We also classify, up to similarity, the linear subspaces of Mn(K)Mn(K) in which every matrix has at most two eigenvalues (respectively, at most one non-zero eigenvalue) in an algebraic closure of KK and which have the maximal dimension among such spaces.