Multiparameter perturbation theory of matrices and linear operators

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.