On the mean density of complex eigenvalues for an ensemble of random matrices with prescribed singular values

Abstract

Given any fixed $N\times N$ positive semi-definite diagonal matrix $G\ge 0$ we derive the explicit formula for the density of complex eigenvalues for random matrices $A$ of the form $A=U\sqrt{G}$} where the random unitary matrices $U$ are distributed on the group $\mathrm{U}(\mathrm{N})$ according to the Haar measure.

10 pages, 1 figure

References in corpus (19)

• Riemannian symmetric superspaces and their origin in random matrix theory
• Statistics of resonance poles, phase shifts and time delays in quantum chaotic scattering for systems with broken time reversal invariance
• Wannier-Stark resonances in optical and semiconductor superlattices
• Truncations of random unitary matrices
• Random Matrices close to Hermitian or unitary: overview of methods and results
• Almost-Hermitian Random Matrices: Eigenvalue Density in the Complex Plane
• Non-Gaussian Non-Hermitean Random Matrix Theory: phase transitions and addition formalism
• Supersymmetry for Systems with Unitary Disorder: Circular Ensembles
• Fractal Weyl laws in discrete models of chaotic scattering
• Quantum-to-classical correspondence in open chaotic systems
• Spectra of Random Contractions and Scattering Theory for Discrete-Time Systems
• Dynamical model for the quantum-to-classical crossover of shot noise
• Quantum Andreev map: A paradigm of quantum chaos in superconductivity
• Fingerprints of classical diffusion in open 2D mesoscopic systems in the metallic regime
• On absolute moments of characteristic polynomials of a certain class of complex random matrices
• "Single Ring Theorem" and the Disk-Annulus Phase Transition
• Resonance distribution in open quantum chaotic systems
• A model for chaotic dielectric microresonators
• Spectra of Random Matrices Close to Unitary and Scattering Theory for Discrete-Time Systems

Cited by in corpus (12)

• Induced Ginibre ensemble of random matrices and quantum operations
• Resonances in open quantum maps
• Exact Relation between Singular Value and Eigenvalue Statistics
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• Lifetime statistics in chaotic dielectric microresonators
• Eigenvalue spectra of asymmetric random matrices for multi-component neural networks
• One-component plasma on a spherical annulus and a random matrix ensemble
• Commutative law for products of infinitely large isotropic random matrices
• The single ring theorem
• Asymptotic mean density of sub-unitary ensemble
• Spectral Properties of Non-Unitary Band Matrices
• Random matrices with prescribed eigenvalues and expectation values for random quantum states