Author:

Keywords:

Science & Technology, Physical Sciences, Mathematics, RANDOM MATRICES, BIORTHOGONAL ENSEMBLES, SINGULAR-VALUES, PRODUCTS, 4904 Pure mathematics

Abstract:

A polynomial ensemble is a probability density function for the position of n real particles of the form 1/Z_n Π(x_k-x_j) det [f_k(x_j)] for certain functions f_1, ..., f_n. Such ensembles appear frequently as the joint eigenvalue density of random matrices. We present a number of transformations that preserve the structure of a polynomial ensemble. These transformations include the restriction of a Hermitian matrix by removing one row and one column, a rank-one modification of a Hermitian matrix, and the extension of a Hermitian matrix by adding an extra row and column with complex Gaussians. A special case of the latter result gives an elementary approach to the joint eigenvalue density of a GUE matrix.