Products of rectangular random matrices: Singular values and progressive scattering

Abstract

We discuss the product of M rectangular random matrices with independent Gaussian entries, which have several applications, including wireless telecommunication and econophysics. For complex matrices an explicit expression for the joint probability density function is obtained using the Harish-Chandra-Itzykson-Zuber integration formula. Explicit expressions for all correlation functions and moments for finite matrix sizes are obtained using a two-matrix model and the method of biorthogonal polynomials. This generalizes the classical result for the so-called Wishart-Laguerre Gaussian unitary ensemble (or chiral unitary ensemble) at M=1, and previous results for the product of square matrices. The correlation functions are given by a determinantal point process, where the kernel can be expressed in terms of Meijer G-functions. We compare the results with numerical simulations and known results for the macroscopic level density in the limit of large matrices. The location of the end points of support for the latter are analyzed in detail for general M. Finally, we consider the so-called ergodic mutual information, which gives an upper bound for the spectral efficiency of a MIMO communication channel with multifold scattering.

Singular Values of Products of Ginibre Random Matrices, Multiple Orthogonal Polynomials and Hard Edge Scaling Limits

“…This was extended by Akemann, Ipsen and Kieburg [4] to the general rectangular case. The determinantal point process is a biorthogonal ensemble [13] with joint probability density function (see [4, formula (18)])…”

“…Very recently, Akemann, Kieburg, and Wei [5] found that the squared singular values of products of complex Ginibre matrices are a determinantal point process on the positive real line. This was further extended to the case of products of rectangular Ginibre matrices by Akemann, Ipsen and Kieburg [4]. The correlation kernels in [2,3,4,5,25] are all expressed in terms of Meijer G-functions.…”

“…Akemann et al [4,5] studied an extension of (1.3) to a two-matrix model and obtained in this framework that for certain polynomials Q k ,…”

“…In [4] it is shown that the biorthogonal functions P k and Q k have integral representations as Meijer G-functions. We rederive these results in Section 3 using only the biorthogonality (1.11).…”

“…. , n, are the squared singular values of Y M , 4) and normalization constant (see [4, formula (21)])…”