# Multiplicative convolution of real asymmetric and real anti-symmetric matrices

- Authors
- Type
- Published Article
- Journal
- Advances in Pure and Applied Mathematics
- Publisher
- De Gruyter
- Publication Date
- Jan 20, 2019
- Volume
- 10
- Issue
- 4
- Pages
- 467–492
- Identifiers
- DOI: 10.1515/apam-2018-0037
- Source
- De Gruyter
- Keywords
- License
- Yellow

## Abstract

The singular values of products of standard complex Gaussian random matrices, or sub-blocks of Haar distributed unitary matrices, have the property that their probability distribution has an explicit, structured form referred to as a polynomial ensemble. It is furthermore the case that the corresponding bi-orthogonal system can be determined in terms of Meijer G-functions, and the correlation kernel given as an explicit double contour integral. It has recently been shown that the Hermitised product X M ⋯ X 2 X 1 A X 1 T X 2 T ⋯ X M T {X_{M}\cdots X_{2}X_{1}AX_{1}^{T}X_{2}^{T}\cdots X_{M}^{T}} , where each X i {X_{i}} is a standard real Gaussian matrix and A is real anti-symmetric, exhibits analogous properties. Here we use the theory of spherical functions and transforms to present a theory which, for even dimensions, includes these properties of the latter product as a special case. As an example we show that the theory also allows for a treatment of this class of Hermitised product when the X i {X_{i}} are chosen as sub-blocks of Haar distributed real orthogonal matrices.