Wigner Matrices and Semicircular Law

  • Zhidong Bai
  • Jack W. Silverstein
Part of the Springer Series in Statistics book series (SSS)


A Wigner matrix is a symmetric (or Hermitian in the complex case) random matrix. Wigner matrices play an important role in nuclear physics and mathematical physics. The reader is referred to Mehta [212] for applications of Wigner matrices to these areas. Here we mention that they also have a strong statistical meaning. Consider the limit of a normalizedWishart matrix. Suppose that x1, …, x n are iid samples drawn from a p-dimensional multivariate normal population N(μ, I p ). Then, the sample covariance matrix is defined as
$$ S_n = \frac{1}{{n - 1}}\sum\limits_{i = 1}^n {(x_i - \bar x)} (x_i - \bar x)',$$
where \(\overline{x}=\frac{1}{n}\sum\nolimits_{i=1}^{n}{x_i}.\) When n tends to infinity \( S_n \rightarrow I_p \) and \(\sqrt {n} (S_n - I_p) \rightarrow \sqrt {p} {W_p}\) It can be seen that the entries above the main diagonal of \(\sqrt {p} {W_p}\) are iid N(0, 1) and the entries on the diagonal are iid N(0, 2). This matrix is called the (standard) Gaussian matrix or Wigner matrix.


Diagonal Element Isomorphism Class Single Edge Isomorphic Graph Gaussian Matrix 
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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.School of Mathematics and Statistics KLAS MOE Northeast Normal UniversityChangchunChina
  2. 2.Department of Statistics and Applied ProbabilityNational University of SingaporeSingaporeSingapore
  3. 3.Department of MathematicsNorth Carolina State UniversityRaleighUS

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