Let 
be a real symmetric
matrix of large order 
having random elements 
that for 
are independently distributed with equal densities,
equal second moments 
, and 
th moments bounded by constants 
independent of 
, 
, and 
. Further, let 
be the number of eigenvalues
of 
that lie in the interval 
for real 
. Then
 |
(Wigner 1955, 1958). This law was first observed by Wigner (1955) for certain special classes of random matrices arising in quantum mechanical investigations.
The distribution of eigenvalues of a symmetric random matrix with entries chosen from a standard
normal distribution is illustrated above for a random 
matrix.
Note that a large real symmetric matrix with random entries taken from a uniform distribution also obeys the semicircle law with the exception that it also possesses exactly one large eigenvalue.
