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Let $A$ a $n \times n$ random matrix with i.i.d $N(0,\sigma^2/n)$ entries. Let $H$ an invertible matrix, and denote $\sigma_H$ the largest singular value of $HAH^{-1}$. My question is : in the large $n$ limit, what is the value of $$\inf_{H\ invertible} \sigma_H$$

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