Operator in a real vector space has an upper block triangular matrix

[vish_maths]

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, let's 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.

 

[fresh_42]

How should this ever be answered without knowing the definition of $T?$ If you want to know whether such a $T$ exists, simply write it down. Otherwise look for possible decompositions like Jordan-Chevalley or Cholesky.