Let $\mathbb K$ be a field. Let $A\in M_n(\mathbb K)$ be the matrix of a semisimple linear operator (that is, $A$ is diagonalizable in the algebraic closure of $\mathbb K$).
Is it true that the centralizer of $A$ in $M_n(\mathbb K)$ can be decomposed into
$C_{M_n({\mathbb K})}(A)= M_{n_1}(\mathbb K)\otimes_{\mathbb K}\ldots\otimes_{\mathbb K} M_{n_r}(\mathbb K), $
where the $n_i$ are the size of the Jordan blocks in the Jordan Canonical Form of $A$ in the algebraic closure of $\mathbb K$?