12 - Matrices with Gaussian Element Densities But with No Unitary or Hermitian Condition Imposed

Abstract

This chapter focuses on matrices with Gaussian element densities but with no unitary or Hermitian condition imposed. An ensemble of matrices whose elements are complex, quaternion, or real numbers, but with no other restrictions as to their Hermitian or unitary character, is of no immediate physical interest, for their eigenvalues may lie anywhere on the complex plane. However, an effort has been made to investigate them and the results are interesting in their own right. To define a matrix ensemble, one has to specify two things: (1) the space T on which the matrices vary and (2) a probability density function over T. With a view to practical applications, one can take T as the set of all real symmetric matrices with a reasonable probability density. For example, one may assume that the matrix elements are independent and have Gaussian probability densities so that all the diagonal elements have the same variance σ1 and the same mean value, whereas all the off-diagonal elements have the mean value zero and the same variance σ2, the ratio of the variances σ1 and σ2 being arbitrary. This case has not yet been considered analytically.