I'm having a hard time putting this into a proof. It's posed as a question in Section 6.2 of Hoffman and Kunze, asking whether it is true or not. It seems obviously true, since if T is left multiplication by A, isn't A just the matrix representation of the transformation? So they would clearly have the same characteristic values? I seem to be missing something. From there I wanted to prove that if A is diagonalizable, this implies L is diagonalizable since they have the same characteristic values.
Thanks sincerely for any clarification you can provide.