Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Multiparameter perturbation theory of matrices and linear operators


Authors: Adam Parusiński and Guillaume Rond
Journal: Trans. Amer. Math. Soc. 373 (2020), 2933-2948
MSC (2010): Primary 47A55; Secondary 13F25, 14P20, 15A18, 26E10
DOI: https://doi.org/10.1090/tran/8061
Published electronically: January 23, 2020
MathSciNet review: 4069237
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract:

We show that a normal matrix $A$ with coefficients in $\mathbb {C}[[X]]$, $X=(X_1, \ldots , X_n)$, can be diagonalized, provided the discriminant $\Delta _A$ of its characteristic polynomial is a monomial times a unit. The proof is an adaptation of our proof of the Abhyankar-Jung Theorem. As a corollary we obtain the singular value decomposition for an arbitrary matrix $A$ with coefficient in $\mathbb {C}[[X]]$ under a similar assumption on $\Delta _{AA^*}$ and $\Delta _{A^*A}$.

We also show real versions of these results, i.e., for coefficients in $\mathbb {R}[[X]]$, and deduce several results on multiparameter perturbation theory for normal matrices with real analytic, quasi-analytic, or Nash coefficients.


References [Enhancements On Off] (What's this?)

References

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 47A55, 13F25, 14P20, 15A18, 26E10

Retrieve articles in all journals with MSC (2010): 47A55, 13F25, 14P20, 15A18, 26E10


Additional Information

Adam Parusiński
Affiliation: Université Côte d’Azur, Université Nice Sophia Antipolis, CNRS, LJAD, Parc Valrose, 06108 Nice Cedex 02, France
Email: adam.parusinski@unice.fr

Guillaume Rond
Affiliation: LASOL, UMI2001, UNAM, Aix Marseille Univ, Mexico
MR Author ID: 759916
Email: guillaume.rond@univ-amu.fr

Received by editor(s): June 18, 2019
Received by editor(s) in revised form: September 24, 2019
Published electronically: January 23, 2020
Additional Notes: The authors’ research was supported in part by ANR project LISA (ANR-17-CE40-0023-03)
Article copyright: © Copyright 2020 American Mathematical Society