We show that a normal matrix A A with coefficients in C [ [ X ] ] \mathbb {C}[[X]] , X = ( X 1 , … , X n ) X=(X_1, \ldots , X_n) , can be diagonalized, provided the discriminant Δ A \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 A with coefficient in C [ [ X ] ] \mathbb {C}[[X]] under a similar assumption on Δ A A ∗ \Delta _{AA^*} and Δ A ∗ A \Delta _{A^*A} . We also show real versions of these results, i.e., for coefficients in R [ [ X ] ] \mathbb {R}[[X]] , and deduce several results on multiparameter perturbation theory for normal matrices with real analytic, quasi-analytic, or Nash coefficients.