Let $A$ be a finite-dimensional complex $C^*$-algebra. Then there exist an integer $r \ge 0$ and integers $n_1,\dots,n_r \in \mathbb N$ such that
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\begin{align*}
A \cong \bigoplus_{j=1}^{r} M_{n_j}(\mathbb C)
\end{align*}
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as $C^*$-algebras, where the empty direct sum for $r=0$ is the zero $C^*$-algebra. If $A\neq 0$, then the multiset $\{n_1,\dots,n_r\}$ is uniquely determined by $A$.