A decomposition result for the Haar distribution on the orthogonal group

TL;DR

This paper provides a new decomposition of the Haar distribution on the orthogonal group, revealing how it can be constructed recursively from lower-dimensional Haar distributions and uniform sphere distributions.

Contribution

It introduces a novel conditional distribution characterization that allows recursive construction of Haar orthogonal matrices from smaller Haar matrices and sphere distributions.

Findings

Conditional distributions involve known probability distributions.

Haar distribution can be built recursively from smaller Haar distributions.

Provides explicit construction method for Haar matrices.

Abstract

Let H be a Haar distributed random matrix on the group of pxp real orthogonal matrices. Partition H into four blocks: (1) the (1,1) element, (2)the rest of the first row, (3) the rest of the first column, and (4)the remaining (p-1)x(p-1) matrix. The marginal distribution of (1) is well known. In this paper, we give the conditional distribution of (2) and (3) given (1), and the conditional distribution of (4) given (1), (2), (3). This conditional specification uniquely determines the Haar distribution. The two conditional distributions involve well known probability distributions namely, the uniform distribution on the unit sphere in p-1 dimensional space and the Haar distribution on (p-2)x(p-2) orthogonal matrices. Our results show how to construct the Haar distribution on pxp orthogonal matrices from the Haar distribution on (p-2)x(p-2) orthogonal matrices coupled with the uniform distribution on the unit sphere in p-1 dimensions.