Problem description:
Let A be an $4\times 4$ complex matrix, determine the largest possible dimension of the subspace $S_A=\{B \in M_n (C)|AB=BA\}$.
My answer is 10. Because all commuting matrices are triangulable, so the largest number of the unit matrices $E_{ij}$ equal to the basis that spans an upper triangular matrix.
Is my answer right? Or can you help to offer a proof.
Thanks.