A Constructive Cayley Representation of Orthogonal Matrices and Applications to Optimization

TL;DR

This paper presents a constructive, efficient algorithm to represent any real orthogonal matrix using the Cayley transform, with applications to improving optimization methods on the orthogonal group.

Contribution

It introduces a numerically efficient algorithm to compute a diagonal signature matrix for Cayley transform representation of orthogonal matrices, with explicit bounds and applications.

Findings

Algorithm requires O(n^3) operations.

Provides explicit bounds on the skew-symmetric generator.

Applications to optimization on the orthogonal group.

Abstract

It is known that every real orthogonal matrix can be brought into the domain of the Cayley transform by multiplication with a suitable diagonal signature matrix. In this paper we provide a constructive and numerically efficient algorithm that, given a real orthogonal matrix $U$, computes a diagonal matrix $D$ with entries in $\{\pm1\}$ such that the Cayley transform of $DU$ is well defined. This yields a representation of $U$ in the form \[ U = D(I-S)(I+S)^{-1}, \] where $S$ is a skew-symmetric matrix. The proposed algorithm requires $O(n^{3})$ arithmetic operations and produces an explicit quantitative bound on the associated skew-symmetric generator. As an application, we show how this construction can be used to control singularities in Cayley-transform-based optimization methods on the orthogonal group.