IMC 1994 · Problem 4

Let $A$ be a $n \times n$ diagonal matrix with characteristic polynomial

$$(x - c_1)^{d_1} (x - c_2)^{d_2} \dots (x - c_k)^{d_k},$$

where $c_1, c_2, \dots, c_k$ are distinct (which means that $c_1$ appears $d_1$ times on the diagonal, $c_2$ appears $d_2$ times on the diagonal, etc. and $d_1 + d_2 + \dots + d_k = n$).

Let $V$ be the space of all $n \times n$ matrices $B$ such that $AB = BA$. Prove that the dimension of $V$ is

$$d_1^2 + d_2^2 + \dots + d_k^2.$$