The key observation of Girko [@G1; @G2] relates the empirical spectral measure of a non-Hermitian matrix to that of a Hermitian matrix. To illustrate the connection, consider the *Cauchy–Stieltjes transform* $s_N$ of the measure $\mu_{{\mathbf}{M}}$, where ${\mathbf}{M}$ is an $N \times N$ matrix, given by $$s_N(z) := \frac{1}{N} \sum_{i=1}^N \frac{1}{ \lambda_i({\mathbf}{M}) - z} = \int_{\mathbb{C}} \frac{1}{x -z } \mu_{{\mathbf}{M}} (dx),$$ for $z \in \mathbb{C}$. Since $s_N$ is analytic everywhere except at the poles (which are exactly the eigenvalues of ${\mathbf}{M}$), the real part of $s_N$ determines the eigenvalues. Let $\sqrt{-1}$ denote the imaginary unit, and set $z = s + \sqrt{-1} t$. Then we can write the real part of $s_N(z)$ as $$\begin{aligned}
    {\operatorname{Re}}( s_N(z)) &= \frac{1}{N} \sum_{i=1}^N \frac{ {\operatorname{Re}}(\lambda_i({\mathbf}{M})) - s }{ \left| \lambda_i({\mathbf}{M}) - z \right|^2} \\
        &= \frac{-1}{2N} \sum_{i=1}^N \frac{\partial}{\partial s} \log \left| \lambda_i({\mathbf}{M}) - z \right|^2 \\
        &= \frac{-1}{2N} \frac{\partial}{\partial s}  \log \det \left( {\mathbf}{M} - z {\mathbf}{I} \right) \left( {\mathbf}{M} - z{\mathbf}{I} \right)^{\ast} \\
        &= -\frac{\partial}{\partial s} \int_{\mathbb{C}} \log x^2  \nu_{{\mathbf}{M} - z {\mathbf}{I}}(dx), \end{aligned}$$ where ${\mathbf}{I}$ denotes the identity matrix. In other words, the task of studying $\mu_{{\mathbf}{M}}$ reduces to studying the measures $\{\nu_{{\mathbf}{M} - z {\mathbf}{I}}\}_{z \in \mathbb{C}}$. The difficulty now is that 