Symmetric Gaussian Matrices
The previous post looked at the distribution of eigenvalues for very general random matrices. In this post, we will look at the eigenvalues of matrices with more structure. Fill an n by n matrix A with values drawn from a standard normal distribution and let M be the average of A and its transpose, i.e. M = ½( A + AT). The eigenvalues will all be real because M is symmetric.
This is called a "Gaussian Orthogonal Ensemble" or GOE. The term is standard but a little misleading because such matrices may not be orthogonal.