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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Multiparameter perturbation theory of matrices and linear operators
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by Adam Parusiński and Guillaume Rond PDF
Trans. Amer. Math. Soc. 373 (2020), 2933-2948 Request permission

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.

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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)
  • © Copyright 2020 American Mathematical Society
  • 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
  • MathSciNet review: 4069237