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!


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).

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

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