Some properties of invariant polynomials with matrix arguments and their applications in econometrics

Chikuse, Yasuko; Davis, A. W.

doi:10.1007/bf02482504

Cited by 38 publications

(77 citation statements)

References 20 publications

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“…In this section, we describe a class of homogeneous polynomials C κ,τ φ (X, Y ) of degrees k and t in the elements of the m × m symmetric complex matrices X and Y , respectively, (see, [6], [7] and [4]). These polynomials are invariant under the simultaneous transformations…”

Section: Invariant Polynomialsmentioning

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: Invariant Polynomialsmentioning

confidence: 99%

Complex random matrices and Rician channel capacity

2005

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“…. , t ) of t, with 1 t i = t; h (j ) (v) is the jth derivative of h with respect to v and the operators , , C , and the notation for the sums are given in [6], see also [4].…”

Section: Theorem 9 (I) the Joint Density Of D And W 1 Ismentioning

confidence: 99%

Singular random matrix decompositions: distributions

Dı́az-Garcı́a

González–Farías

2005

Journal of Multivariate Analysis

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Assuming that Y has a singular matrix variate elliptically contoured distribution with respect to the Hausdorff measure, the distributions of several matrices associated to QR, modified QR, SV and polar decompositions of matrix Y are determined, for central and non-central, non-singular and singular cases, as well as their relationship to the Wishart and pseudo-Wishart generalized singular and non-singular distributions. Some of these results are also applied to two particular subfamilies of elliptical distributions, the singular matrix variate normal distribution and the singular matrix variate symmetric Pearson type VII distribution.

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“…For example, the noncentral distributions were found using zonal polynomials or the hypergeometric function with matrix argument, [15,17]. Double noncentral distributions and distributions associated with eigenvalues of some specific matrices, were solved through the application of a generalization of zonal polynomials called invariant polynomials with matrix argument, [3,4]. A problem that has not been solved completely is the one related to the distribution of random singular matrices, which are not unusual to find in practical and theoretical problems.…”

Section: Introductionmentioning

confidence: 99%