  Let $0 \in \Gamma$ and $\Gamma \setminus \{0\}$ be a subgroup of the complex
numbers of unit modulus. Define $\mathcal{H}_{n}(\Gamma)$ to be the set of all
$n\times n$ Hermitian matrices with entries in $\Gamma$, whose diagonal entries
are zero. The matrices $A,B\in \mathcal{H}_{n}(\Gamma)$ are said to be
switching equivalent if there is a diagonal matrix $D$, in which the diagonal
entries belong to $\Gamma \setminus \{0\}$, such that $D^{-1} A D=B$. We find a
characterization, in terms of fundamental cycles of graphs, of switching
equivalence of matrices in $\mathcal{H}_{n}(\Gamma)$. We give sufficient
conditions to characterize the cospectral matrices in
$\mathcal{H}_{n}(\Gamma)$. We find bounds on the number of switching
equivalence classes of all mixed graphs with the same underlying graph. We also
provide the size of all switching equivalence classes of mixed cycles, and give
a formula that calculates the size of a switching equivalence class of a mixed
plane graph. We also discuss an action of the automorphism group of a graph on
switching equivalence classes of matrices in $\mathcal{H}_{n}(\Gamma)$.
