A Wigner matrix in random matrix theory is a Hermitian matrix 
whose entries strictly above the diagonal
are independent and identically
distributed, with mean zero and a common finite variance.
The diagonal entries are identically distributed real
random variables with a fixed law independent
of 
and finite mean
and variance, independent
of one another and of the other entries. The lower entries are determined by 
, where the bar denotes
the complex conjugate. Real symmetric
matrices are an important special case (Tao 2012).
A common convention gives the off-diagonal entries variance 1 and studies the scaled matrix 
, equivalently using entry variance 
in the normalized matrix.
These random matrices are the setting of Wigner's
semicircle law.
This usage is distinct from the Wigner D-matrix, a group representation parametrized by Euler angles.
