Multiparameter perturbation theory of matrices and linear operators

Abstract

We show that a normal matrix A with coefficient in C[[X]], X = (X1, . . . , Xn), can be diagonalized, provided the discriminant ∆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 C[[X]] under a similar assumption on ∆AA * and ∆A * A.We also show real versions of these results, i.e. for coefficients in R[[X]], and deduce several results on multiparameter perturbation theory for normal matrices with real analytic, quasi-analytic, or Nash coefficients.