# Exact relation between singular value and eigenvalue statistics

Kieburg M, Kösters H (2016)
RANDOM MATRICES-THEORY AND APPLICATIONS 5(4): 1650015.

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Journal Article | Original Article | Published | English

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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.
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Kieburg M, Kösters H. Exact relation between singular value and eigenvalue statistics. RANDOM MATRICES-THEORY AND APPLICATIONS. 2016;5(4): 1650015.
Kieburg, M., & Kösters, H. (2016). Exact relation between singular value and eigenvalue statistics. RANDOM MATRICES-THEORY AND APPLICATIONS, 5(4), 1650015. doi:10.1142/S2010326316500155
Kieburg, M., and Kösters, H. (2016). Exact relation between singular value and eigenvalue statistics. RANDOM MATRICES-THEORY AND APPLICATIONS 5:1650015.
Kieburg, M., & Kösters, H., 2016. Exact relation between singular value and eigenvalue statistics. RANDOM MATRICES-THEORY AND APPLICATIONS, 5(4): 1650015.
M. Kieburg and H. Kösters, “Exact relation between singular value and eigenvalue statistics”, RANDOM MATRICES-THEORY AND APPLICATIONS, vol. 5, 2016, : 1650015.
Kieburg, M., Kösters, H.: Exact relation between singular value and eigenvalue statistics. RANDOM MATRICES-THEORY AND APPLICATIONS. 5, : 1650015 (2016).
Kieburg, Mario, and Kösters, Holger. “Exact relation between singular value and eigenvalue statistics”. RANDOM MATRICES-THEORY AND APPLICATIONS 5.4 (2016): 1650015.
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