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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?

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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).

So dependent on the precision of the calculation, this will tell you with what probability you will not be able to distinguish two nearly identical eigenvalues.

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  • $\begingroup$ Can we assert anything about non symmetric matrix? $\endgroup$
    – FourierFlux
    Commented Mar 30, 2021 at 21:29
  • $\begingroup$ random non-symmetric matrices form the Ginibre ensemble, for which the nearest eigenvalue spacing distribution vanishes as $s^3$ for small spacings, hence $P(f)\propto f^4$. $\endgroup$
    – Carlo Beenakker
    Commented Mar 31, 2021 at 6:30

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