A commuting family {"Ai"} of matrices can be simultaneously triangularized, i.e. there exists a unitary matrix "Q" such that, for every "Ai" in the given family, "Q Ai Q*" is upper triangular. This can be readily deduced from the above proof. Take element "A" from {"Ai"} and again consider an eigenspace "VA". Then "VA" is invariant under all matrices in {"Ai"}. Therefore, all matrices in {"Ai"} must share one common eigenvector in "VA". Induction then proves the claim. As a corollary, we have that every commuting family of normal matrices can be simultaneously diagonalized.