Some Combinatorial Aspects of the Spectra of Normally Distributed Random Matrices

Let U be a real n x n matrix whose entries uij are random variables, and let A and B be fixed n x n real symmetric matrices. Statisticians (e.g., see [OU]) have been interested in the distribution of the eigenvalues d l , . . . , O n of the matrix AUBUt, where denotes transpose. Of particular interest are the quantities tr((AU = C 0; for k = 1,2,. . . , since these determine the eigenvalues. More generally, one may consider the distribution of arbitrary symmetric functions of the eigenvalues of AUBU1. Thus let us regard any symmetric polynomial f (say with real coefficients) in the variables X I , . . . , x, as a function on n x n matrices by defining f (U) to be the value off a t the eigenvalues of the matrix U. In these terms, we have