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.

中文翻译：

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具有Rodrigues公式的对偶复矩阵的因式分解和广义根

本文基于De Moivre公式的广泛推广，提供了一种有效的方法来获取对偶复（2×2）矩阵的幂和根。我们还使用极坐标分解和\（\ mathfrak {so} _4 \）的直接和结构来解决普通\（3×3 \）和\（4×4 \）矩阵的情况。为有理数次幂推导的紧凑显式表达式正式地（具有周期性损失）扩展为实数，复数或什至对偶数，这允许在那些情况下定义矩阵的某些超越函数类，而无需参考无穷级数或可替代地，获得总和这些系列中的一个（可以在文本中找到明显的示例）。此外，我们建议采用分解程序\（\ mathrm {M}（n，{\ mathbb {C}} [\ varepsilon]）\），\（nle 4 \）基于极性分解和作者最近在实际中提出的广义Euler型过程案件。我们的方法使用双重双四元数及其投影形式，在欧几里得语环境中称为Rodrigues的向量。对某些子代数的限制会在各个领域产生有趣的应用，例如在经典力学和机器人学中广泛使用的螺杆几何形状，相对论和电动力学中的洛伦兹群的复杂表示形式，计算机视觉中的保形映射，散射过程的物理学等等。在这里，我们仅对这些主题提供简短的评论，并提供一些明确的示例来说明该方法。