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

B. V. Verbitskii

Sbornik. Mathematics, Tome 20 (1973) no. 1, pp. 53-65

Résumé

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 $0$. 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.