A Constructive Cayley Representation of Orthogonal Matrices and Applications to Optimization

Iwo Biborski

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arXiv · 2601.16271

A Constructive Cayley Representation of Orthogonal Matrices and Applications to Optimization

Iwo Biborski

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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.

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Iwo Biborski. 2026-01-22. A Constructive Cayley Representation of Orthogonal Matrices and Applications to Optimization. https://arxiv.org/abs/2601.16271

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