In this paper we prove the following assertion. Let $A(x)$ be an $n\times n$ matrix whose elements belong to $C^k[0,b]$ , where $k\geqslant0$ and $0<b<\infty$ . Furthermore, let $\{\sigma_j(x)\}_1^m$ ( $m\leqslant n$ ) be the distinct eigenvalues of $A(x)$ belonging to $C^k[0,b]$ . Then, if $A(x)$ for all $x\in[0,b]$ is similar to a Jordan matrix $J(x)$ , in which to each eigenvalue $\sigma_j(x)$ there corresponds a constant number of Jordan blocks whose dimension is also independent of $x\in[0,b]$ , it follows that $A(x)$ is smoothly similar to $J(x)$ on $[0,b]$ . Bibliography: 6 titles.