Hello All, I was trying to prove that an operator T in a real vector space V has an upper block triangular matrix with each block being 1 X 1 or 2 X 2 and without using induction. The procedure which i followed was : We already know that an operator in a real vector space has either a one dimensional invariant subspace or a 2 dimensional invariant subspace. Whatever be the case now, lets begin with the vector(s) which span these subspaces. Let U denote this subspace ----- (1) Now, if i am able to prove that there exists an another subspace W such that T is an invariant operator on the direct sum of U and W , then we can prove that operator T in a real vector space V has an upper block triangular matrix . I need a direction on proving the latter part. I sincerely thank you for the help.