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Mathematics   > Home   > Advances in Pure and Applied Mathematics   > Issue 1   > Article

Products of Complex Rectangular and Hermitian Random Matrices

Produits de Complexes Rectangulaires et Matrices aléatoires hermitiennes


Mario Kieburg
The University of Melbourne
Australia



Published on 3 July 2020   DOI : 10.21494/ISTE.OP.2020.0543

Abstract

Résumé

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Very recently, it has been shown for products of real matrices with anti-symmetric matrices of even dimension that the traditional harmonic analysis on matrix groups developed by Harish-Chandra et al. needs to be modified when considering the group action on general symmetric spaces of matrices. In the present work, we consider the product of complex random matrices with Hermitian matrices, in particular the former can be also rectangular while the latter has not to be positive definite and is considered as a fixed matrix as well as a random matrix. This generalises an approach for products involving the Gaussian unitary ensemble (GUE) and circumvents the use there of non-compact group integrals. We derive the joint probability density function of the real eigenvalues and, additionally, prove transformation formulas for the bi-orthogonal functions and kernels.

Very recently, it has been shown for products of real matrices with anti-symmetric matrices of even dimension that the traditional harmonic analysis on matrix groups developed by Harish-Chandra et al. needs to be modified when considering the group action on general symmetric spaces of matrices. In the present work, we consider the product of complex random matrices with Hermitian matrices, in particular the former can be also rectangular while the latter has not to be positive definite and is considered as a fixed matrix as well as a random matrix. This generalises an approach for products involving the Gaussian unitary ensemble (GUE) and circumvents the use there of non-compact group integrals. We derive the joint probability density function of the real eigenvalues and, additionally, prove transformation formulas for the bi-orthogonal functions and kernels.

products of independent random matrices polynomial ensemble multiplicative convolution Pólya frequency functions spherical transform bi-orthogonal ensembles

products of independent random matrices polynomial ensemble multiplicative convolution Pólya frequency functions spherical transform bi-orthogonal ensembles