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.