# Probability finite precision random matrix has distinct eigenvalues

copied from math stack exchange

There is a theorem which says the probability/size of a random matrix having repeated eigenvalues is 0 and this result is used in many fields. What I am wondering is, how does this translate to real systems with finite precision?

How can we extend this result to make some statement on the probability a random matrix with finite precision has no repeat eigenvalues? Obviously it's no longer 0 but how do we quantify it?

The probability $$P(f)$$ that the spacing $$s$$ of two eigenvalues of a random real symmetric matrix is smaller than the average spacing $$\bar{s}$$ by a fraction $$f$$ is given by $$P(f)=1-e^{-\pi f^2/4},$$ according to the Wigner surmise (valid for a large-dimensional matrix under very general conditions).
• random non-symmetric matrices form the Ginibre ensemble, for which the nearest eigenvalue spacing distribution vanishes as $s^3$ for small spacings, hence $P(f)\propto f^4$. Mar 31 at 6:30