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Hello everybody, here is my question:

Assume A is a random symmetric $nxn$ matrix whose entries are independent, normally distributed with mean zero and variance 2 on the diagonal and 1 off diagonal (to my knowledge such a random matrix is said to belong to the 'Gaussian orthogonal ensemble'). The joint pdf for the eigenvalues of A is well known, but i was wondering:

does there exist a precise formula for the probability that all the eigenvalues of A have norm greater then epsilon?

Equivalently which is the probability that the least singular value of A is smaller than epsilon?

I am computing the intrinsic volume of singular symmetric matrices of (Frobenius or trace-square) norm one and I need this precise formula to perform the epsilon limit using tubes. I am sorry if this is a well known result, but I was not able to find it in the literature. In case I would really appreciate a reference for this.

Thanks everyone!

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I'm not sure offhand (and don't have time to check at the moment) if the GOE version of this is known, but the distribution least singular value of a nonsymmetric $n \times n$ matrix with i.i.d. normal entries was determined exactly by Edelman in this paper (may be behind a pay-wall).

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  • $\begingroup$ I don't see an immediate way to relate this to the GOE case... :) $\endgroup$
    – A. Lerario
    Commented Mar 1, 2012 at 1:44
  • $\begingroup$ Neither do I, but perhaps Edelman's proof can be adapted. $\endgroup$
    – Mark Meckes
    Commented Mar 1, 2012 at 12:04

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