  We consider $n\times n$ non-Hermitian random matrices with independent
entries and a variance profile, as well as an additive deterministic diagonal
deformation. We show that the support of the asymptotic eigenvalue distribution
in the complex plane exactly coincides with the $\varepsilon$-pseudospectrum in
the consecutive limits $n \to \infty$ and $\varepsilon \to 0$. Furthermore, we
provide a description of this support in terms of a single real-valued function
on the complex plane. As a level set of this locally real analytic function,
the spectral edge is a real analytic variety of dimension at most one.
