The eigenvalues of random symmetric matrices

Abstract

LetA=(a ij ) be ann ×n matrix whose entries forij are independent random variables anda ji =a ij . Suppose that everya ij is bounded and for everyi>j we haveEa ij =μ,D 2 a ij 2 andEa ii =v.

E. P. Wigner determined the asymptotic behavior of the eigenvalues ofA (semi-circle law). In particular, for anyc>2σ with probability 1-o(1) all eigenvalues except for at mosto(n) lie in the intervalI=(−cn,cn).

We show that with probability 1-o(1)all eigenvalues belong to the above intervalI if μ=0, while in case μ>0 only the largest eigenvalue λ1 is outsideI, and

$$\lambda _1 = \frac{{\Sigma _{i,j} a_{ij} }}{n} + \frac{{\sigma ^2 }}{\mu } + O\left( {\frac{I}{{\sqrt n }}} \right)$$

i.e. λ1 asymptotically has a normal distribution with expectation (n−1)μ+v+(σ2/μ) and variance 2σ2 (bounded variance!).

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Füredi, Z., Komlós, J. The eigenvalues of random symmetric matrices. Combinatorica 1, 233–241 (1981). https://doi.org/10.1007/BF02579329

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AMS subject classification (1980)

  • 15 A 52
  • 15 A 18
  • 05 C 50