Spaces of matrices with a sole eigenvalue

Let K be an arbitrary (commutative) field and L be an algebraic closure of it. Let V be a linear subspace of M_n(K), with n>2. We show that if every matrix of V has at most one eigenvalue in K, then dim V<=1+n(n-1)/2. If every matrix of V has a sole eigenvalue in L and dim V=1+n(n-1)/2, we show that V is similar to the space of all upper-triangular matrices with equal diagonal entries, except if n=3 and K has characteristic 3, or if n=4 and K has characteristic 2. In both of those special cases, we classify the exceptional solutions up to similarity.