# Products of Rectangular Random Matrices: Singular Values and Progressive Scattering

Akemann G, Ipsen J, Kieburg M (2013) *Physical Review E* 88(5): 052118.

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Abstract

We discuss the product of $M$ rectangular random matrices with independentGaussian entries, which have several applications including wirelesstelecommunication and econophysics. For complex matrices an explicit expressionfor the joint probability density function is obtained using theHarish-Chandra--Itzykson--Zuber integration formula. Explicit expressions forall correlation functions and moments for finite matrix sizes are obtainedusing a two-matrix model and the method of bi-orthogonal polynomials. Thisgeneralises the classical result for the so-called Wishart--Laguerre Gaussianunitary ensemble (or chiral unitary ensemble) at M=1, and previous results forthe product of square matrices. The correlation functions are given by adeterminantal point process, where the kernel can be expressed in terms ofMeijer $G$-functions. We compare the results with numerical simulations andknown results for the macroscopic density in the limit of large matrices. Thelocation of the endpoints of support for the latter are analysed in detail forgeneral $M$. Finally, we consider the so-called ergodic mutual information,which gives an upper bound for the spectral efficiency of a MIMO communicationchannel with multi-fold scattering.

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Akemann G, Ipsen J, Kieburg M. Products of Rectangular Random Matrices: Singular Values and Progressive Scattering.

*Physical Review E*. 2013;88(5):052118.Akemann, G., Ipsen, J., & Kieburg, M. (2013). Products of Rectangular Random Matrices: Singular Values and Progressive Scattering.

*Physical Review E*,*88*(5), 052118. doi:10.1103/PhysRevE.88.052118Akemann, G., Ipsen, J., and Kieburg, M. (2013). Products of Rectangular Random Matrices: Singular Values and Progressive Scattering.

*Physical Review E*88, 052118.Akemann, G., Ipsen, J., & Kieburg, M., 2013. Products of Rectangular Random Matrices: Singular Values and Progressive Scattering.

*Physical Review E*, 88(5), p 052118. G. Akemann, J. Ipsen, and M. Kieburg, “Products of Rectangular Random Matrices: Singular Values and Progressive Scattering”,

*Physical Review E*, vol. 88, 2013, pp. 052118. Akemann, G., Ipsen, J., Kieburg, M.: Products of Rectangular Random Matrices: Singular Values and Progressive Scattering. Physical Review E. 88, 052118 (2013).

Akemann, Gernot, Ipsen, Jesper, and Kieburg, Mario. “Products of Rectangular Random Matrices: Singular Values and Progressive Scattering”.

*Physical Review E*88.5 (2013): 052118.
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PMID: 24329225

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arXiv 1307.7560