Given a matrix $A\in\mathbb{C}^{n\times n}$, we denote by symbol $\mathrm{In}_d (A):= (n_<,n_>,n_1)$, the discrete inertia (or inertia w.r.t. the unit circle) of $A$, where $n_<,n_>$ and $n_1$ indicate the number of eigenvalues of $A$ with modulus smaller than 1, greater than 1 and equal to 1, respectively.
By the well-known Sylvester's Law of Inertia, the "standard" inertia of an Hermitian matrix (i.e. the number of positive, negative and zero eigenvalues) is invariant under congruence transformations. Now, is it possible to characterize which kind of transformations leave the discrete inertia of $A$ invariant?
References addressing this problem or related ones will be appreciated. Thanks in advance.