Multiparameter perturbation theory of matrices and linear operators

Abstract

We show that a normal matrix $A$ with coefficient in $\mathbb{C}[[X]]$, $X=(X_1,\dots ,X_n)$, 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 ${\Delta }_{A{A}^{\ast }}$ and ${\Delta }_{A^{\ast }A}$. 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.

15 pages - to appear in Trans. Amr. Math. Soc