TY - JOUR
AB - We use classical results from harmonic analysis on matrix spaces to investigate the relation between the joint densities of the singular values and the eigenvalues for complex random matrices which are bi-unitarily invariant (also known as isotropic or unitary rotation invariant). We prove that each of these joint densities determines the other one. Moreover, we construct an explicit formula relating both joint densities at finite matrix dimension. This relation covers probability densities as well as signed densities. With the help of this relation we derive general analytical relations among the corresponding kernels and biorthogonal functions for a specific class of polynomial ensembles. Furthermore, we show how to generalize the relation between the singular value and eigenvalue statistics to certain situations when the ensemble is deformed by a term which breaks the bi-unitary invariance.
AU - Kieburg, Mario
AU - Kösters, Holger
ID - 2907442
IS - 4
JF - RANDOM MATRICES-THEORY AND APPLICATIONS
KW - Bi-unitarily invariant complex random matrix ensembles
KW - singular value
KW - densities
KW - eigenvalue densities
KW - spherical function
KW - spherical
KW - transform
KW - determinantal point processes
SN - 2010-3263
TI - Exact relation between singular value and eigenvalue statistics
VL - 5
ER -