the fixed point equation. Because of the dependencies between entries in the matrix ${\mathbf}Z_N$, directly computing the trace of the resolvent of the Hermitization of ${\mathbf}Z_N$ is troublesome. To circumvent this issue, in Section \[sec:resolvent\], instead of taking the trace of the resolvent, we instead take the partial trace and consider a $2m \times 2m$ matrix-valued Stieltjes transform. Then we show this partial trace approximately satisfies a matrix-valued fixed point equation.

In Section \[sec:singularvalue\], we deduce a bound for the least singular value of the matrix $\frac{1}{\sqrt{N}}({\mathbf}{Y}_N + {\mathbf}{A}_N) - z {\mathbf}{I}$ from the known bounds on the least singular values of the individual matrices ${\mathbf}Y_{N,k} +{\mathbf}A_{N,k}$. We finally complete the proof of Theorem \[thm:circular\] in Section \[sec:complete\].

The proof of Theorem \[thm:prodWigner\] is very similar to the proof of Theorem \[thm:main\]. In fact, there are only a few places in the proof of Theorem \[thm:main\] where the condition $|\rho_k| < 1$ is required. We prove Theorem \[thm:prodWigner\] in Section \[sec:Wigner\].

A remark from free probability
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Notation
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We use asymptotic notation (such as $O,o,\Omega$) under the assumption that $N \rightarrow \infty$. We use $X \ll Y, Y \gg X, Y=\Omega(X)$, or $X = O(Y)$ to denote the bound 