A Constructive Cayley Representation of Orthogonal Matrices and Applications to Optimization

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 $\{\pm 1\}$ 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.