show that Theorem \[thm:main\] is a short corollary of Theorem \[thm:circular\]. This same linearization trick was used in [@OS] to study products of non-Hermitian matrices with iid entries. Similar techniques were also used in [@A; @HT] to study general self-adjoint polynomials of self-adjoint random matrices.

Sections \[sec:resolvent\], \[sec:singularvalue\], and \[sec:complete\] are dedicated to proving Theorem \[thm:circular\]. Following the ideas of Girko [@G1; @G2], we compute the limiting spectral measure of a non-Hermitian random matrix ${\mathbf}{M}$, by employing the method of *Hermitizaition*. Given an $N \times N$ matrix ${\mathbf}{M}$, we recall that the empirical spectral measure of ${\mathbf}{M}$ is given by $$\mu_{\mathbf}{M} := \frac{1}{N} \sum_{i=1}^N \delta_{\lambda_i({\mathbf}{M})},$$ where $\lambda_1({\mathbf}{M}), \ldots, \lambda_N({\mathbf}{M})$ are the eigenvalues of ${\mathbf}{M}$. We let $\nu_{{\mathbf}{M}}$ denote the symmetric empirical measure built from the singular values of ${\mathbf}{M}$. That is, $$\nu_{{\mathbf}{M}} := \frac{1}{2N} \sum_{i =1}^N \left( \delta_{\sigma_i({\mathbf}{M})} + \delta_{-\sigma_i({\mathbf}{M})} \right),$$ where $\sigma_1({\mathbf}{M}) \geq \cdots \geq \sigma_N({\mathbf}{M}) \geq 0$ are the singular values of ${\mathbf}{M}$. In particular, $$\sigma_1({\mathbf}{M}) := \sup_{\|x\| = 1} \| {\mathbf}{M} x \|$$ is the largest singular value of ${\mathbf}{M}$ and $$\sigma_N({\mathbf}{M}) := \inf_{\|x\| = 1} \| {\mathbf}{M} x \|$$ is the smallest singular value, both of which will play a key role in our analysis below.

