Sbornik. Mathematics, Tome 20 (1973) no. 1, pp. 53-65
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B. V. Verbitskii. On a global property of a matrix-valued function of one variable. Sbornik. Mathematics, Tome 20 (1973) no. 1, pp. 53-65. http://geodesic.mathdoc.fr/item/SM_1973_20_1_a2/
@article{SM_1973_20_1_a2,
author = {B. V. Verbitskii},
title = {On~a~global property of a~matrix-valued function of one variable},
journal = {Sbornik. Mathematics},
pages = {53--65},
year = {1973},
volume = {20},
number = {1},
language = {en},
url = {http://geodesic.mathdoc.fr/item/SM_1973_20_1_a2/}
}
TY - JOUR
AU - B. V. Verbitskii
TI - On a global property of a matrix-valued function of one variable
JO - Sbornik. Mathematics
PY - 1973
SP - 53
EP - 65
VL - 20
IS - 1
UR - http://geodesic.mathdoc.fr/item/SM_1973_20_1_a2/
LA - en
ID - SM_1973_20_1_a2
ER -
%0 Journal Article
%A B. V. Verbitskii
%T On a global property of a matrix-valued function of one variable
%J Sbornik. Mathematics
%D 1973
%P 53-65
%V 20
%N 1
%U http://geodesic.mathdoc.fr/item/SM_1973_20_1_a2/
%G en
%F SM_1973_20_1_a2
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.
[2] W. Wasow, “On holomorphically similar matrices”, J. math. analysis, 4:2 (1962), 202–206 | DOI | MR | Zbl
[3] Y. Sibuya, “Some global properties of matrices of function of one variable”, Math. Ann., 161:1 (1965), 67–77 | DOI | MR | Zbl
[4] P.-F. Hsich, J. Sibuya, “A global analysis of matrices of function of several variables”, J. math. analysis, 14:2 (1966), 332–340 | DOI | MR
[5] S. Khabbaz, G. Stengle, “An application $K$-theory to the global analysis of matrix valued function”, Math. Ann., 179:2 (1969), 115–122 | DOI | MR | Zbl
[6] R. Narasimkhan, Analiz na deistvitelnykh i kompleksnykh mnogoobraziyakh, izd-vo «Mir», Moskva, 1971
