  We introduce the volume-preserving equivalence among symmetric matrix-valued
map-germs which is the unimodular version of Bruce's $\mathcal{G}$-equivalence.
The key concept to deduce unimodular classification out of classification
relative to $\mathcal{G}$-equivalence is symmetrical quasi-homogeneity, which
is a generalization of the condition for a $2 \times 2$ symmetric matrix-valued
map-germ in Corollary~2.1 (ii) by Bruce, Goryunov and Zakalyukin. If a
$\mathcal{G}$-equivalence class contains a symmetrically quasi-homogeneous
representative, the class coincides with that relative to the volume-preserving
equivalence (up to orientation reversing diffeomorphism in case if the ground
field is real). By using that we show that all the simple classes relative to
$\mathcal{G}$-equivalence in Bruce's list coincides with those relative to the
volume preserving equivalence. Then, we classify map-germs from the plane to
the set of $2 \times 2$ and $3 \times 3$ real symmetric matrices of corank at
most $1$ and of $\mathcal{G}_e$-codimension less than $9$ and we show some of
the normal forms split into two different unimodular singularities. We provide
several examples to illustrate that non simplicity does not imply non
symmetrical quasi-homogeneity and the condition that a map-germ is
symmetrically quasi-homogeneous is stronger than one that each component of the
map-germ is quasi-homogeneous. We also present an example of non symmetrically
quasi-homogeneous normal form relative to $\mathcal{G}$ and its corresponding
formal unimodular normal form.
