A Constructive Cayley Representation of Orthogonal Matrices and Applications to Optimization

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.