Diffeomorphic Logarithm of Special Orthogonal Matrices

Abstract

The special orthogonal group $\mathbb{SO}_n$ is a Lie group whose geometry and local structure are encoded by the exponential map in its Lie algebra $\mathbf{Skew}_n$, the set of skew-symmetric matrices. The associated multi-valued inverse problem -- the matrix logarithm -- in $\mathbb{SO}_n$ exhibits a highly nontrivial local diffeomorphism structure, which differs from the matrix logarithm for invertible matrices. This work characterizes the local diffeomorphism structure of the exponential in the set of skew-symmetric matrices where its derivative is invertible. We show that this set with an invertible derivative can be organized into diffeomorphic regions, using a canonical alignment of Schur decompositions. In particular, the region that contains the principal logarithm has a special multiplicity structure: each matrix in $\mathbb{SO}_n$ admits at most two skew-symmetric preimages in this region. Based on this geometric framework, we introduce the diffeomorphic logarithm of special orthogonal matrices together with an efficient and stable algorithm. Moreover, it is applied to the Karcher mean problem in $\mathbb{SO}_n$, demonstrating continuous behavior of the mean under perturbations of the data, which is not captured by the principal logarithm.