Thread: Linear Operator

Let T be a linear operator on an n-dimensional space and suppose that T has n distinct characteristics values. Prove that any linear operator which commutes with T is a polynomial in T.

Originally Posted by problem

Let T be a linear operator on an n-dimensional space and suppose that T has n distinct characteristics values. Prove that any linear operator which commutes with T is a polynomial in T.

Start with the fact that T has n distinct characteristics values. What can you say about such an operator (think in terms of matrices)? What can you say about the matrix of an operator that commutes with T?