Invariant polynomials with two matrix arguments extending the zonal polynomials: Applications to multivariate distribution theory

Davis, A. W.

doi:10.1007/bf02480302

Cited by 133 publications

(170 citation statements)

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“…However, the eigenvalue densities of complex noncentral Wishart matrices cannot be solved in terms of complex zonal polynomials. Here we derive these densities using invariant polynomials, which are proposed by Davis [6], [7]. These invariant polynomials have two matrix arguments, which extend the single matrix argument of zonal polynomials.…”

Section: Introductionmentioning

confidence: 99%

Complex random matrices and Rician channel capacity

2005

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The 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. The connection between the complex Wishart matrix theory and information theory is given. This facilitates the evaluation of the most important information-theoretic measure, the so-called ergodic channel capacity. In particular, the capacity of multipleinput, multiple-output (MIMO) Rician distributed channels is investigated. We consider both spatially correlated and uncorrelated MIMO Rician channels and derive the 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. Key words. complex random matrix, complex noncentral Wishart matrix, zonal polynomial, invariant polynomial, Rician distributed MIMO channel, ergodic channel capacity AMS(MOS) subject classification. 94A15, 94A05, 60E05, 62H10To appear in Problems of Information Transmission. RésuméOnétudie la densité des valeurs propres de matrices de Wishart noncentrales complexes en vue d'applicationsà un problème ouvert de la théorie de l'information. On obtient la densité conjointe de la plus grande et de la plus petite de ces valeurs propres au moyen de polynômes zonaux complexes et de polynômes invariants. Onétablie la relation entre ces matrices et la théorie de l'information pourévaluer la très importante capacité ergodique d'un canal. Onétudie la capacité des canaux riciens distribuésà entrées et sorties multiples (MIMO). On considère les canaux correlés et non correlés dans l'espace et l'on obtient facilement la borne supérieure exacte des capacités ergodiques. Les résul-tats numériques montrent que la corrélation du canal réduit la capacité du système de communication.

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Section: Introductionmentioning

confidence: 99%

Complex random matrices and Rician channel capacity

2005

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“…However, it is now clear that this same recursion applies for any generating functions D(t) and P (t) related by (5).…”

Section: Generating Functions With a Single Variablementioning

confidence: 83%

Generating functions and short recursions, with applications to the moments of quadratic forms in noncentral normal vectors

Hillier¹,

Kan²,

Wang³

2008

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“…For properties and applications of invariant polynomials we refer to Davis [6,7], Chikuse [4] and Nagar and Gupta [16]. Let κ, λ, φ and ρ be ordered partitions of the nonnegative integers k, , f = k + and r [5] Matrix-variate Gauss hypergeometric distribution 339 respectively into not more than m parts.…”

Section: Some Well-known Results and Definitionsmentioning

confidence: 99%

Matrix-Variate Gauss Hypergeometric Distribution

Gupta

Nagar

2012

J. Aust. Math. Soc.

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In this paper, we propose a matrix-variate generalization of the Gauss hypergeometric distribution and study several of its properties. We also derive probability density functions of the product of two independent random matrices when one of them is Gauss hypergeometric. These densities are expressed in terms of Appell's first hypergeometric function F 1 and Humbert's confluent hypergeometric function Φ 1 of matrix arguments.

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Invariant polynomials with two matrix arguments extending the zonal polynomials: Applications to multivariate distribution theory

Cited by 133 publications

References 13 publications

Complex random matrices and Rician channel capacity

Complex random matrices and Rician channel capacity

Generating functions and short recursions, with applications to the moments of quadratic forms in noncentral normal vectors

Matrix-Variate Gauss Hypergeometric Distribution

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