The Probability that a Random Real Gaussian Matrix haskReal Eigenvalues, Related Distributions, and the Circular Law

@article{Edelman1997ThePT,
  title={The Probability that a Random Real Gaussian Matrix haskReal Eigenvalues, Related Distributions, and the Circular Law},
  author={Alan Edelman},
  journal={Journal of Multivariate Analysis},
  year={1997},
  volume={60},
  pages={203-232}
}
  • A. Edelman
  • Published 1 February 1997
  • Mathematics
  • Journal of Multivariate Analysis
LetAbe annbynmatrix whose elements are independent random variables with standard normal distributions. Girko's (more general) circular law states that the distribution of appropriately normalized eigenvalues is asymptotically uniform in the unit disk in the complex plane. We derive the exact expected empirical spectral distribution of the complex eigenvalues for finiten, from which convergence in the expected distribution to the circular law for normally distributed matrices may be derived… 

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References

SHOWING 1-10 OF 18 REFERENCES
How many eigenvalues of a random matrix are real
Let A be an n x n matrix whose elements are independent random variables with standard normal distributions. As n oo , the expected number of real eigenvalues is asymptotic to V/7r . We obtain a
Statistical Ensembles of Complex, Quaternion, and Real Matrices
Statistical ensembles of complex, quaternion, and real matrices with Gaussian probability distribution, are studied. We determine the over‐all eigenvalue distribution in these three cases (in the
Eigenvalue statistics of random real matrices.
  • Lehmann, Sommers
  • Computer Science, Mathematics
    Physical review letters
  • 1991
TLDR
The joint probability density of eigenvalues in a Gaussian ensemble of real asymmetric matrices, which is invariant under orthogonal transformations is determined, which indicates thatrices of the type considered appear in models for neural-network dynamics and dissipative quantum dynamics.
A Canonical Form for Real Matrices under Orthogonal Transformations.
  • F. Murnaghan, A. Wintner
  • Mathematics
    Proceedings of the National Academy of Sciences of the United States of America
  • 1931
If A is a square matrix of order n with real or complex elements it is well known that it may be reduced by means of a unitary transformation U to a matrix of the same order all of whose elements
Limiting behavior of the norm of products of random matrices and two problems of Geman-Hwang
Abstract : In the theory of large random matrices, how to dominate the norm of a random matrix is a very important problem. This paper considers a different type of random matrices, namely -W to the
On the Distribution of a Scaled Condition
In this note, we give the exact distribution of a scaled condition number used by Demmel to model the probability that matrix inversion is diicult. Speciically, consider a random matrix A and the
On the distribution of a scaled condition number
In this note, we give the exact distribution of a scaled condition number used by Demmel to model the probability that matrix inversion is difficult. Specifically, consider a random matrix A and the
Aspects of Multivariate Statistical Theory
  • R. Muirhead
  • Mathematics
    Wiley Series in Probability and Statistics
  • 1982
Tables. Commonly Used Notation. 1. The Multivariate Normal and Related Distributions. 2. Jacobians, Exterior Products, Kronecker Products, and Related Topics. 3. Samples from a Multivariate Normal
Eigenvalues and condition numbers of random matrices
Given a random matrix, what condition number should be expected? This paper presents a proof that for real or complex $n \times n$ matrices with elements from a standard normal distribution, the ex...
THE SPECTRAL RADIUS OF LARGE RANDOM MATRICES
Soit {n ij }, i=1,2... j=1,2... une variable aleatoire i.i.d. avec Em 11 =0 et Em 11 2 =σ 2 . Pour chaque n, on definit M n ={m ij } (1≤i,j≤n) la matrice n×n dont la composante (i, j) est m ij . On
...
...