Spectral density of a Wishart model for nonsymmetric correlation matrices

Spectral domain of large nonsymmetric correlated Wishart matrices

We study complex eigenvalues of the Wishart model for nonsymmetric correlation matrices. The model is defined for two statistically equivalent but different Gaussian real matrices, as C=AB(t)/T, where B(t) is the transpose of B and both matrices A and B are of dimensions N×T. If A and B are uncorrelated, or equivalently if C vanishes on average, it is known that at large matrix dimension the domain of the eigenvalues of C is a circle centered-at-origin and the eigenvalue density depends only on the radial distances. We consider actual correlation in A and B and derive a result for the contour describing the domain of the bulk of the eigenvalues of C in the limit of large N and T where the ratio N/T is finite. In particular, we show that the eigenvalue domain is sensitive to the correlations. For example, when C is diagonal on average with the same element c≠0, the contour is no longer a circle centered at origin but a shifted ellipse. In this case we explicitly derive a result for the spectral density which again depends only on the radial distances. For more general cases, we show that the contour depends on the symmetric and antisymmetric parts of the correlation matrix resulting from the ensemble-averaged C. If the correlation matrix is normal, then the contour depends only on its spectrum. We also provide numerics to justify our analytics.

Spectral density of the noncentral correlated Wishart ensembles

Wishart ensembles of random matrix theory have been useful in modeling positive definite matrices encountered in classical and quantum chaotic systems. We consider nonzero means for the entries of the constituting matrix A which defines the correlated Wishart matrix as W=AA(†), and refer to the ensemble of such Wishart matrices as the noncentral correlated Wishart ensemble (nc-CWE). We derive the Pastur self-consistent equation which describes the spectral density of nc-CWE at large matrix dimension.

Products of rectangular random matrices: singular values and progressive scattering

We discuss the product of M rectangular random matrices with independent Gaussian entries, which have several applications, including wireless telecommunication and econophysics. For complex matrices an explicit expression for the joint probability density function is obtained using the Harish-Chandra-Itzykson-Zuber integration formula. Explicit expressions for all correlation functions and moments for finite matrix sizes are obtained using a two-matrix model and the method of biorthogonal polynomials. This generalizes the classical result for the so-called Wishart-Laguerre Gaussian unitary ensemble (or chiral unitary ensemble) at M=1, and previous results for the product of square matrices. The correlation functions are given by a determinantal point process, where the kernel can be expressed in terms of Meijer G-functions. We compare the results with numerical simulations and known results for the macroscopic level density in the limit of large matrices. The location of the end points of support for the latter are analyzed in detail for general M. Finally, we consider the so-called ergodic mutual information, which gives an upper bound for the spectral efficiency of a MIMO communication channel with multifold scattering.