Yes. Sorry about that -- as I was falling asleep I realized I'd completely overlooked the question of independence, but didn't have the energy to get out of bed and post that . . . --Dan On 2012-11-22, at 6:38 AM, Veit Elser wrote:
I have to agree with Brent, since any reasonable measure should respect the symmetry M -> -M (and matrices with at least one 0 eigenvalue should constitute a set of measure 0 among the Hermitians).
Nice!
--Dan
That only shows the probability of 5 negative and 9 positive eigenvalues is the same as 9 negative and 5 positive. The claim 2^(-N) assumes much more, that the eigenvalues are independently distributed. In fact, they are very far from independent, making the actual probability much smaller: c^(-N^2). This result is asymptotic, for large N (the limit of interest in statistical mechanics). The number c is well known, but it is not 2.