Factorization and Generalized Roots of Dual Complex Matrices with Rodrigues’ Formula

Abstract

The paper provides an efficient method for obtaining powers and roots of dual complex $2\times 2$ matrices based on a far reaching generalization of De Moivre’s formula. We also resolve the case of normal $3\times 3$ and $4\times 4$ matrices using polar decomposition and the direct sum structure of $\mathfrak {so}_4$. The compact explicit expressions derived for rational powers formally extend (with loss of periodicity) to real, complex or even dual ones, which allows for defining some classes of transcendent functions of matrices in those cases without referring to infinite series or alternatively, obtain the sum of those series (explicit examples may be found in the text). Moreover, we suggest a factorization procedure for $\mathrm {M}(n,{\mathbb {C}}[\varepsilon ])$, $n\le 4$ based on polar decomposition and generalized Euler type procedures recently proposed by the author in the real case. Our approach uses dual biquaternions and their projective version referred to in the Euclidean setting as Rodrigues’ vectors. Restrictions to certain subalgebras yield interesting applications in various fields, such as screw geometry extensively used in classical mechanics and robotics, complex representations of the Lorentz group in relativity and electrodynamics, conformal mappings in computer vision, the physics of scattering processes and probably many others. Here we only provide brief comments on these subjects with several explicit examples to illustrate the method.