Let $A$ be an $n\times n$ matrix with complex entries, and let $P(x)$ be the characteristic polynomial of $A$. Then $P(x)$ can be written as $(x-\lambda_1)^{k_1}(x-\lambda_2)^{k_2}\cdots(x-\lambda_\ell)^{k_\ell}$, where $\lambda_i\not=\lambda_j$ if $i\not=j$, and $k_i\geq 1$ for $i=1,2,\dots,\ell$. Each $\lambda_i$ is an eigenvalue of $A$ with (algebraic) multiplicity $k_i$. If the dimension of the eigenspace of $\lambda_i$ is $k_i$ for all $i$, then $A$ can be written as a diagonal matrix, and vice versa.