Exact relation between singular value and eigenvalue statistics
Kieburg, Mario
Kösters, Holger
Bi-unitarily invariant complex random matrix ensembles
singular value
densities
eigenvalue densities
spherical function
spherical
transform
determinantal point processes
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.
World Sci Publ Co Inc
2016
info:eu-repo/semantics/article
doc-type:article
text
https://pub.uni-bielefeld.de/record/2907442
Kieburg M, Kösters H. Exact relation between singular value and eigenvalue statistics. <em>RANDOM MATRICES-THEORY AND APPLICATIONS</em>. 2016;5(4): 1650015.
eng
info:eu-repo/semantics/altIdentifier/doi/10.1142/S2010326316500155
info:eu-repo/semantics/altIdentifier/issn/2010-3263
info:eu-repo/semantics/altIdentifier/issn/2010-3271
info:eu-repo/semantics/altIdentifier/wos/000388112000004
info:eu-repo/semantics/closedAccess