\right)\end{aligned}$$ for $A$ sufficiently large.

We now recall that ${\mathbf}{Y}_{N,1}$ and ${\mathbf}{Y}_{N,2}$ are independent, and, on the event $\Omega_N$, the entries of $\left( \frac{1}{\sqrt{N}} {\mathbf}{Y}_{N,2} \right)^{-1}$ are bounded in magnitude by $N^{A'}$. In addition, we observe that $z^2 \left( \frac{1}{\sqrt{N}} {\mathbf}{Y}_{N,2} \right)^{-1}$ is a complex symmetric matrix since ${\mathbf}{Y}_{N,2}$ is a real symmetric matrix. Therefore, by Theorem \[thm:NgLSV\], there exists $A>0$ such that $${\mathbb{P}}\left( \left\| \left( \frac{1}{\sqrt{N}} {\mathbf}{Y}_{N,1} - z^2 \left( \frac{1}{\sqrt{N}} {\mathbf}{Y}_{N,2}\right)^{-1} \right)^{-1} \right\| \geq N^{A/2} \bigg| \Omega_N \right) = O(N^{-2}).$$ This verifies , and hence the proof of Theorem \[thm:least-sing-valueWigner\] is complete.

Truncation {#sec:truncationproof}
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