On a global property of a matrix-valued function of one variable

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 $[0,b]$.
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