Hello I am studying for the qualifying exam in algebra and I am having trouble solving this seemingly easy problem. If A is a matrix whose minimal polynomial and characteristic polynomial agree, and B commutes with A then B is a polynomial in A.
I have shown that the dimension of the subspace of polynomials in A must be equal to the dimension of the underlying vector space. Clearly this subspace is contained in the subspace of matrices that commute with A. So if I can show the latter must have dimension less than or equal to the dimension of V, I'll be done. But I don't see how to show that.
Or is there an easier way?