Exponentials of skew-symmetric matrices and logarithms of orthogonal matrices

Two widely used methods for computing matrix exponentials and matrix logarithms are, respectively, the scaling and squaring and the inverse scaling and squaring. Both methods become effective when combined with Padé approximation. This paper deals with the computation of exponentials of skew-symmetric matrices and logarithms of orthogonal matrices. Our main goal is to improve these two methods by exploiting the special structure of skew-symmetric and orthogonal matrices. Geometric features of the matrix exponential and logarithm and extensions to the special Euclidean group of rigid motions are also addressed.