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Given $A$, a $n\times n$ random matrix with centred Gaussian (real) i.i.d entries with variance $\sigma$, what is the variance in the spectral norm $\left(\sqrt{\text{Largest eigenvalue of } A^\dagger A}\right)$ of $A$?

If $n$ is large, can the central limit theorem be used? If yes, how could one obtain the variance of the distribution of the spectral norm?

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www-math.mit.edu/~edelman/homepage/papers/Eig.pdf might be of interest to you –  Evan Sep 6 '13 at 17:15

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