Joint Eigenvalue Density Research Articles

Rank One Non-Hermitian Perturbations of Hermitian beta -Ensembles of Random Matrices

We provide a tridiagonal matrix model and compute the joint eigenvalue density of a rank one non-Hermitian perturbation of a random matrix from the Gaussian or Laguerre beta -ensemble.

Cutset Bounds on the Capacity of MIMO Relay Channels

We analyze the ergodic capacity of multiple-input multiple-output (MIMO) Rayleigh-fading relay channels. We first derive the probability density function of a sum of independent complex central Wishart matrices—called the central hyper-Wishart matrix —and its joint eigenvalue density. We then derive a trace representation for the max-flow min-cut upper bound on the ergodic capacity of general full-duplex MIMO relay channels where each communicating node is equipped with $ N$ antennas and has access only to respective receive channel state information. We also establish the Schur monotonicity theorem for this cutset bound as a functional of the signal-to-noise ratios (SNRs) of three communication links. We further characterize the exact ergodic capacity in the regularity SNR regime where the upper and lower bounds coincide.

Limits for circular Jacobi beta-ensembles

Bourgade, Nikeghbali and Rouault recently proposed a matrix model for the circular Jacobi β-ensemble. This is a generalization of the Dyson circular β-ensemble but equipped with an additional parameter b controlling the order of a spectrum singularity. We calculate the scaling limits for expected products of characteristic polynomials of circular Jacobi β-ensembles. For a fixed constant b, the resulting limit near the spectrum singularity is proven to be a new multivariate function. When b=βNd/2, the scaling limits in the bulk and at the soft edge agree with those of the Hermite (Gaussian), Laguerre (chiral) and Jacobi β-ensembles proved in Desrosiers and Liu (2014). As corollaries, for even β the scaling limits of point correlation functions for the ensemble are given. Besides, a transition from the spectrum singularity to the soft edge limit is observed as b goes to infinity. The positivity of two special multivariate hypergeometric functions, which appear as one factor of the joint eigenvalue densities for Jacobi/Wishart β-ensembles with general covariance and Gaussian β-ensembles with source, will also be shown.

The Real Ginibre Ensemble with $$k=O(n)$$ k = O ( n ) Real Eigenvalues

We consider the ensemble of real Ginibre matrices conditioned to have positive fraction $$\alpha >0$$ of real eigenvalues. We demonstrate a large deviations principle for the joint eigenvalue density of such matrices and introduce a two phase log-gas whose stationary distribution coincides with the spectral measure of the ensemble. Using these tools we provide an asymptotic expansion for the probability $$p^n_{\alpha n}$$ that an $$n\times n$$ Ginibre matrix has $$k=\alpha n$$ real eigenvalues and we characterize the spectral measures of these matrices.

Random matrix ensembles involving Gaussian Wigner and Wishart matrices, and biorthogonal structure.

We consider four nontrivial ensembles involving Gaussian Wigner and Wishart matrices. These are relevant to problems ranging from multiantenna communication to random supergravity. We derive the matrix probability density, as well as the eigenvalue densities for these ensembles. In all cases the joint eigenvalue density exhibits a biorthogonal structure. A determinantal representation, based on a generalization of Andréief's integration formula, is used to compactly express the r-point correlation function of eigenvalues. This representation circumvents the complications encountered in the usual approaches, and the answer is obtained immediately by examining the joint density of eigenvalues. We validate our analytical results using Monte Carlo simulations.

Diversity‐multiplexing trade‐off of linear dispersion coded multi‐input–multi‐output systems

In this study, the authors study the diversity-multiplexing trade-off (DMT) of multi-input–multi-output (MIMO) systems in which linear dispersion (LD) codes are used. They first focus on an LD-coded 2 × 2 MIMO systems at the high signal-to-noise ratio (SNR) regime. By using the joint eigenvalue density, the outage probability is presented in a very compact expression, which establishes the DMT framework at this regime. Simulation results show that the derived expression for the outage probability is sufficiently accurate at high SNR regimes but less accurate in the transition region. Finally, for the special case of multi-input–single-output, they derive both the outage probability and the DMT framework inexact and closed form formulas at all SNRs, including finite SNRs.

Joint density of eigenvalues in spiked multivariate models.

The classical methods of multivariate analysis are based on the eigenvalues of one or two sample covariance matrices. In many applications of these methods, for example to high dimensional data, it is natural to consider alternative hypotheses which are a low rank departure from the null hypothesis. For rank one alternatives, this note provides a representation for the joint eigenvalue density in terms of a single contour integral. This will be of use for deriving approximate distributions for likelihood ratios and 'linear' statistics used in testing.

The beta-Wishart ensemble

We introduce a “broken-arrow” matrix model for the β-Wishart ensemble, which improves on the traditional bidiagonal model by generalizing to non-identity covariance parameters. We prove that its joint eigenvalue density involves the correct hypergeometric function of two matrix arguments, and a continuous parameter β > 0. If we choose β = 1, 2, 4, we recover the classical Wishart ensembles of general covariance over the reals, complexes, and quaternions. Jack polynomials are often defined as the eigenfunctions of the Laplace-Beltrami operator. We prove that Jack polynomials are in addition eigenfunctions of an integral operator defined as an average over a β-dependent measure on the sphere. When combined with an identity due to Stanley, we derive a definition of Jack polynomials. An efficient numerical algorithm is also presented for simulations. The algorithm makes use of secular equation software for broken arrow matrices currently unavailable in the popular technical computing languages. The simulations are matched against the cdfs for the extreme eigenvalues. The techniques here suggest that arrow and broken arrow matrices can play an important role in theoretical and computational random matrix theory including the study of corners processes. We provide a number of simulations illustrating the extreme eigenvalue distributions that are likely to be useful for applications. We also compare the n → ∞ answer for all β with the free-probability prediction.

Local Eigenvalue Density for General MANOVA Matrices

We consider random n\times n matrices of the form (XX*+YY*)^{-1/2}YY*(XX*+YY*)^{-1/2}, where X and Y have independent entries with zero mean and variance one. These matrices are the natural generalization of the Gaussian case, which are known as MANOVA matrices and which have joint eigenvalue density given by the third classical ensemble, the Jacobi ensemble. We show that, away from the spectral edge, the eigenvalue density converges to the limiting density of the Jacobi ensemble even on the shortest possible scales of order 1/n (up to \log n factors). This result is the analogue of the local Wigner semicircle law and the local Marchenko-Pastur law for general MANOVA matrices.

Asymptotic power of sphericity tests for high-dimensional data

This paper studies the asymptotic power of tests of sphericity against perturbations in a single unknown direction as both the dimensionality of the data and the number of observations go to infinity. We establish the convergence, under the null hypothesis and contiguous alternatives, of the log ratio of the joint densities of the sample covariance eigenvalues to a Gaussian process indexed by the norm of the perturbation. When the perturbation norm is larger than the phase transition threshold studied in Baik, Ben Arous and Péché [Ann. Probab. 33 (2005) 1643–1697] the limiting process is degenerate, and discrimination between the null and the alternative is asymptotically certain. When the norm is below the threshold, the limiting process is nondegenerate, and the joint eigenvalue densities under the null and alternative hypotheses are mutually contiguous. Using the asymptotic theory of statistical experiments, we obtain asymptotic power envelopes and derive the asymptotic power for various sphericity tests in the contiguity region. In particular, we show that the asymptotic power of the Tracy–Widom-type tests is trivial (i.e., equals the asymptotic size), whereas that of the eigenvalue-based likelihood ratio test is strictly larger than the size, and close to the power envelope.

Complex bimatrix variate generalised beta distributions

In this paper, the study of bivariate generalised beta types I and II distributions is extended to the complex matrix variate case, for which the corresponding density functions are found. In addition, for complex bimatrix variate beta type I distributions, several basic properties, including the joint eigenvalue density and the maximum eigenvalue distribution, are studied.

On the MIMO Channel Capacity for the Nakagami-$m$ Channel

This paper presents the multiple-input multiple-output (MIMO) channel capacity over the Nakagami-m fading channel. The joint eigenvalue density function of W = HH' where H is the channel matrix, is derived in closed form for H (2 times 2) and any integer values of m, as well as for H (2 times 3) with to = 2 and to = 3. The marginal eigenvalue distribution of W is also derived in a closed-form expression. All the results are validated by numerical Monte Carlo simulations and are in excellent agreement.

Quadratic Forms on Complex Random Matrices and Multiple-Antenna Systems

In this correspondence, the densities of quadratic forms on complex random matrices and their joint eigenvalue densities are derived for applications to information theory. These densities are represented by complex hypergeometric functions of matrix arguments, which can be expressed in terms of complex zonal polynomials. The derived densities are used to evaluate the two most important information-theoretic measures, the so-called ergodic channel capacity and capacity versus outage of multiple-input multiple-output (MIMO) spatially correlated Rayleigh-distributed wireless communication channels. We also derive the probability density function of the mutual information between transmitted and received complex signals of MIMO systems with a finite number of transmit and receive antennas. Numerical results show how channel correlation degrades the capacity of MIMO communication systems.

Complex singular wishart matrices and applications

In this paper, complex singular Wishart matrices and their applications to information theory are investigated. In particular, a volume element on the space of positive semidefinite m × m Hermitian matrices of rank n < m is introduced and some transformation properties are established. The Jacobian for the change of variables in the singular value decomposition of general m × n complex matrices is derived. Then, the density functions are formulated for all rank n complex singular Wishart distributions. From this, the joint eigenvalue density of low rank complex Wishart matrices are derived. The derived densities are used to evaluate the most important information-theoretic measure, the so-called ergodic channel capacity of multiple-input multiple-output (MIMO) spatially correlated Rayleigh distributed wireless communication channels.

Complex random matrices and Rician channel capacity

Eigenvalue densities of complex noncentral Wishart matrices are investigated to study an open problem in information theory. Specifically, the largest, smallest, and joint eigenvalue densities of complex noncentral Wishart matrices are derived. These densities are expressed in terms of complex zonal polynomials and invariant polynomials. A connection between the complex Wishart matrix theory and information theory is given. This facilitates evaluation of the most important information-theoretic measure, the so-called ergodic channel capacity. In particular, the capacity of multiple-input multiple-output (MIMO) Rician distributed channels is investigated. We consider both spatially correlated and uncorrelated MIMO Rician channels and derive exact and easily computable tight upper bound formulas for ergodic capacities. Numerical results are also given, which show how the channel correlation degrades the capacity of the communication system.