On Absolute Moments of Characteristic Polynomials of a Certain Class of Complex Random Matrices

The integer moments of the spectral determinant | det (zI − W) |² of complex random matrices W are obtained in terms of the characteristic polynomial of the positive-semidefinite matrix WW ^† for the class of matrices W = AU, where A is a given matrix and U is random unitary. This work is motivated by studies of complex eigenvalues of random matrices and potential applications of the obtained results are discussed in this context. The research in Nottingham was supported by EPSRC grant EP/C515056/1: “Random Matrices and Polynomials: a tool to understand complexity”. Part of this work was carried out during the Newton Institute programme on Random Matrix Approaches in Number Theory.