The sum of independent Wishart matrices, taken from distributions with unequal covariance matrices, plays a crucial role in multivariate statistics, and has applications in the fields of quantitative finance and telecommunication. However, analytical results concerning the corresponding eigenvalue statistics have remained unavailable, even for the sum of two Wishart matrices. This can be attributed to the complicated and rotationally noninvariant nature of the matrix distribution that makes extracting the information about eigenvalues a nontrivial task. Using a generalization of the Harish-Chandra-Itzykson-Zuber integral, we find exact solution to this problem for the complex Wishart case when one of the covariance matrices is proportional to the identity matrix, while the other is arbitrary. We derive exact and compact expressions for the joint probability density and marginal density of eigenvalues. The analytical results are compared with numerical simulations and we find perfect agreement. © EPLA, 2014.

Santosh Kumar

Physics

(SoNS) School of Natural Sciences

Fulltext

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In this work, we consider the weighted difference of two independent complex Wishart matrices and derive the joint probability density function of the corresponding eigenvalues in a finite-dimension scenario using two distinct approaches. The first derivation involves the use of unitary group integral, while the second one relies on applying the derivative principle. The latter relates the joint probability density of eigenvalues of a matrix drawn from a unitarily invariant ensemble to the joint probability density of its diagonal elements. Exact closed form expressions for an arbitrary order correlation function are also obtained and spectral densities are contrasted with Monte Carlo simulation results. Analytical results for moments as well as probabilities quantifying positivity aspects of the spectrum are also derived. Additionally, we provide a large-dimension asymptotic result for the spectral density using the Stieltjes transform approach for algebraic random matrices. Finally, we point out the relationship of these results with the corresponding results for difference of two random density matrices and obtain some explicit and closed form expressions for the spectral density and absolute mean. © 2020 IOP Publishing Ltd.

Journal of Physics A: Mathematical and Theoretical

Spectral statistics for the difference of two Wishart matrices

We comment on the paper 'Cutset Bounds on the Capacity of MIMO Relay Channels' by Jeong et al. and point out that, unlike what appears from a remark and some other contents by these authors, the matrix distribution for the sum of two complex random Wishart matrices has already been derived by Kumar for the general case of arbitrary covariance matrices and not only for the special case when one of them is assumed proportional to the identity matrix. The latter assumption has been made only for deriving the corresponding eigenvalue distribution. Furthermore, we draw attention to the result that when all covariance matrices are chosen proportional to the identity matrix, then it is possible to obtain exact and closed form expressions for the sum of an arbitrary number of Wishart matrices and not only for two as considered by Jeong et al. © 2013 IEEE.

IEEE Access

Comments on "cutset Bounds on the Capacity of MIMO Relay Channels"

The sum of Wishart matrices has an important role in multiple-input multiple-output (MIMO) multiple access channel (MAC) and MIMO relay channel. In this paper, we present a new closed-form expression for the marginal density of one of the unordered eigenvalues of a sum of $K$ complex central Wishart matrices having covariance matrices proportional to the identity matrix. The expression is general and allows for any set of linear coefficients. The derived expression is used to obtain the ergodic sum-rate capacity for the MIMO-MAC and MIMO relay cases, both as closed-form expressions. We also present a very simple expression to approximate the sum of Wishart matrices by one equivalent Wishart matrix. The agreement between the exact eigenvalue distribution and numerical simulations is perfect, whereas for the approximate solution the difference is indistinguishable. © 1967-2012 IEEE.

IEEE Transactions on Vehicular Technology

On the Exact and Approximate Eigenvalue Distribution for Sum of Wishart Matrices

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. © 2015 American Physical Society.