The eigenvalues of random symmetric matrices


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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  1. [1]

    L. Arnold, On the asymptotic distribution of the eigenvalues of random matrices,J. Math. Analysis and Appl. 20 (1967), 262–268.

    MATH  Article  Google Scholar 

  2. [2]

    F. R. Grantmacher,Applications of the theory of matrices, Intersciences, 1959.

  3. [3]

    U. Grenander,Probabilities on algebraic structures, Almquist and Wiksell, Stockholm 1963, 178–180.

    Google Scholar 

  4. [4]

    F. Juhász, On the spectrum of a random graph, in:Algebraic methods in graph theory (Lovász et al., eds),Coll. Math. Soc. J. Bolyai 25, North-Holland 1981, 313–316.

    Google Scholar 

  5. [5]

    L. Lovász,Combinatorial problems and exercises, Akadémiai Kiadó-North-Holland, Budapest-Amsterdam, 1979.

    Google Scholar 

  6. [6]

    A. Ralston,A first course in numerical analysis, McGraw-Hill, 1965.

  7. [7]

    A. Rényi,Foundations of probability, Holden-Day, San Francisco, 1970.

    Google Scholar 

  8. [8]

    E. P. Wigner, Characteristic vectors of bordered matrices with infinite dimensions,Ann. Math. 62 (1955), 548–564.

    Article  MathSciNet  Google Scholar 

  9. [9]

    E. P. Wigner, On the distribution of the roots of certain symmetric matrices,Ann. Math. 67 (1958), 325–327.

    Article  MathSciNet  Google Scholar 

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Füredi, Z., Komlós, J. The eigenvalues of random symmetric matrices. Combinatorica 1, 233–241 (1981).

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AMS subject classification (1980)

  • 15 A 52
  • 15 A 18
  • 05 C 50