Multiparameter perturbation theory of matrices and linear operators

TL;DR

This paper develops a multiparameter perturbation theory for matrices and linear operators, showing diagonalization conditions for matrices with coefficients in power series rings and deriving singular value decompositions under specific discriminant conditions.

Contribution

It extends diagonalization and singular value decomposition results to matrices with power series coefficients, under conditions on the discriminant, using an adaptation of Abhyankar-Jung Theorem.

Findings

Normal matrices can be diagonalized when discriminant is a monomial times a unit.

Singular value decomposition is obtained under similar discriminant conditions.

Results apply to real analytic, quasi-analytic, or Nash coefficients in multiparameter perturbation theory.

Abstract

We show that a normal matrix $A$ with coefficient 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 the algorithm of proof of 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.