On absolute moments of characteristic polynomials of a certain class of complex random matrices

TL;DR

This paper derives formulas for the moments of spectral determinants of certain complex random matrices, linking them to the characteristic polynomial of related Hermitian matrices, with implications for understanding complex eigenvalues.

Contribution

It provides explicit expressions for moments of spectral determinants of matrices of the form W=AUU, connecting them to Hermitian matrix polynomials, a novel approach in this context.

Findings

Explicit formulas for spectral determinant moments of W=AUU matrices.

Connection established between spectral determinants and Hermitian matrix characteristic polynomials.

Potential applications in studying complex eigenvalues of random matrices.

Abstract

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 Hermitian 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 in this context are discussed.