Multiplicative Convolution of Real Asymmetric and Real Antisymmetric Matrices

TL;DR

This paper develops a theoretical framework using spherical functions to analyze the singular value distributions of multiplicative products involving real asymmetric and antisymmetric matrices, extending known results to new matrix classes.

Contribution

It introduces a unified theory for the singular value distributions of Hermitised products involving real matrices, including antisymmetric and sub-blocks of Haar orthogonal matrices.

Findings

Explicit structured distribution forms for these matrix products.

Bi-orthogonal systems expressed via Meijer G-functions.

Extension of properties to sub-blocks of Haar orthogonal matrices.

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 $X_M \cdots X_2 X_1A X_1^T X_2^T \cdots X_M^T$, where each $X_i$ is a standard real complex Gaussian matrix, and $A$ is real anti-symmetric shares 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 $X_i$ are chosen as sub-blocks of Haar distributed real orthogonal matrices.