${\mathbf}{F}_{N,k}$ be complex symmetric [@NgPC]. In particular, the proof only uses that the $(i,j)$-entry of ${\mathbf}{F}_{N,k}$ be equal to the $(j,i)$-entry. We present the most general version below.

\[thm:NgLSV\] For each $k=1,2$, let ${\mathbf}{F}_{N,k}$ be a $N \times N$ complex symmetric matrix whose entries are bounded in magnitude by $N^\alpha$, for some $\alpha > 0$. Then, under the assumptions of Theorem \[thm:prodWigner\], for any $B > 0$, there exists $A>0$ (depending on $\alpha, B$) such that $${\mathbb{P}}\left( \min_{k=1,2} \sigma_{N}( {\mathbf}{Y}_{N,k} + {\mathbf}{F}_{N,k}) \leq N^{-A} \right) = O(N^{-B}).$$

We now prove Theorem \[thm:least-sing-valueWigner\]

Let $A$ be a large positive constant to be chosen later. By the Borel-Cantelli lemma, it suffices to show that, for almost every $z \in \mathbb{C}$, $${\mathbb{P}}\left( \left\| ({\mathbf}{Z}_N - z {\mathbf}{I})^{-1} \right\| \geq N^{A} \right) = O(N^{-2}).$$ By , we observe that, for $z \neq 0$, $({\mathbf}{Z}_N - z {\mathbf}{I})^{-1}$ has the form $$\begin{bmatrix} z \left(\frac{1}{N} {\mathbf}{Y}_{N,1} {\mathbf}{Y}_{N,2} - z^2 {\mathbf}{I} \right)^{-1} & \left(\frac{1}{N} {\mathbf}{Y}_{N,1} {\mathbf}{Y}_{N,2} - z^2 {\mathbf}{I} \right)^{-1} \frac{1}{\sqrt{N}} {\mathbf}{Y}_{N,1} \\ z^2 \left(\frac{1}{N} {\mathbf}{Y}_{N,2} {\mathbf}{Y}_{N,1} - z^2 {\mathbf}{I} \right)^{-1} \frac{1}{\sqrt{N}} {\mathbf}{Y}_{N,2} & z \left( \frac{1}{N} {\mathbf}{Y}_{N,2} {\mathbf}{Y}_{N,1} - z^2 {\mathbf}{I} \right)^{-1} \end{bmatrix}$$ provided the relevant inverses exist. Thus, it suffices to show that the spectral 