If A is an $n \times n$ matrix over the field $F$ with characteristic polynomial $f = (x - C_1)^{d_1} \dots (x - C_k)^{d_k}$

Question

If A is an $n \times n$ matrix over the field $F$ with characteristic polynomial

$$f = (x - c_1)^{d_1} \dots (x - c_k)^{d_k}.$$

What is the trace of $A$?

My attempt: $A$ is similar to an matrix of Jordan $J$ that has on the diagonal all eigenvalues ​​$c_i$ with multiplicities $d_i$. Then the Matrix of Jordan $J$ has a trace $\sum_{i=1} ^{k} c_i d_i$. Now knowing that similar matrices have the same trace, I conclude that the trace of $A$ is $\sum_{i=1} ^{k} c_i d_i$. Is it correct?