Exact Relation between Singular Value and Eigenvalue Statistics

Abstract

We use classical results from harmonic analysis on matrix spaces to investigate the relation between the joint density of the singular values and of the eigenvalues of complex random matrices which are bi-unitarily invariant (also known as isotropic or unitary rotation invariant). We prove that one 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 eigenvalue and singular value statistics to certain situations when the ensemble is deformed by a term which breaks the bi-unitary invariance.

46 pages; minor revision with a few corrections and simplifications

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• Polynomial Ensembles and Pólya Frequency Functions
• Additive Matrix Convolutions of Pólya Ensembles and Polynomial Ensembles
• Asymptotic behavior of the multiplicative counterpart of the Harish-Chandra integral and the $S$-transform
• Co-rank 1 projections and the randomised Horn problem
• Spectral statistics for the difference of two Wishart matrices
• Derivative principles for invariant ensembles
• Products of Complex Rectangular and Hermitian Random Matrices
• Products of random matrices from fixed trace and induced Ginibre ensembles
• Complete diagrammatics of the single ring theorem
• Averages of products and ratios of characteristic polynomials in polynomial ensembles
• Cyclic Pólya Ensembles on the Unitary Matrices and their Spectral Statistics
• Hard Edge Statistics of Products of Pólya Ensembles and Shifted GUE's
• On the extreme value statistics of normal random matrices and 2D Coulomb gases: Universality and finite N corrections
• Relations between generalised Wishart matrices, the Muttalib--Borodin model and matrix spherical functions
• Rate of convergence at the hard edge for various Pólya ensembles of positive definite matrices
• Hard edge asymptotics of correlation functions between singular values and eigenvalues
• Average mutual information for random fermionic Gaussian quantum states
• Correlation functions between singular values and eigenvalues