A general mathematical formulation of the n × n proper orthogonal matrix, that corresponds to a rigid rotation in n -dimensional real Euclidean space, is given here. It is shown that a rigid rotation depends on an angle (principal angle) and on a set of (n ‒ 2) principal axes. The latter, however, can be more conveniently replaced by only two orthogonal directions that identify the plane of rotation. The inverse problem, that is, how to compute these principal rotation parameters from the rotation matrix, is also treated. In this paper, the Euler Theorem is extended to rotations in n -dimensional spaces by a constructive proof that establishes the relationship between orientation of the displaced orthogonal axes in n dimensions and a minimum sequence of rigid rotations. This fundamental relationship, which introduces a new decomposition for proper orthogonal matrices (those identifying an orientation), can be expressed either by a product or a sum of the same rotation matrices. A similar decomposition in terms of the skew-symmetric matrices is also given. The extension of the rigid rotation formulation to n -dimensional complex Euclidean spaces, is also provided. Finally, we introduce the Ortho-Skew real matrices, which are simultaneously proper orthogonal and skew-symmetric and which exist in even dimensional spaces only, and the Ortho-Skew-Hermitian complex matrices which are orthogonal and Skew-Hermitian. The Ortho-Skew and the Ortho-Skew-Hermitian matrices represent the extension of the scalar imaginary to the matrix field.

中文翻译：

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关于 n 维空间中的刚性旋转概念

这里给出 n × n 固有正交矩阵的一般数学公式 ，它对应于 n 维实欧几里得空间中的刚性旋转 。结果表明，刚性旋转取决于角度（主角）和一组 （n ‒ 2）主轴。然而，后者可以更方便地仅由识别旋转平面的两个正交方向代替。还解决了反问题，即如何从旋转矩阵计算这些 主要旋转参数 。在本文中，将欧拉定理扩展到 n中的 旋转 构造空间证明三维空间，该证明建立了位移正交轴在 n个 维度上的方向与刚性旋转的最小序列之间的关系 。这种基本关系为适当的正交矩阵（识别方向的矩阵）引入了新的分解，可以用 相同旋转矩阵的 乘积或总和表示。 还给出了关于斜对称矩阵的类似分解。还提供了将刚性旋转公式扩展到 n 维复欧几里德空间的方法。最后，我们介绍 Ortho-Skew 实数矩阵，它们同时具有正交性和对称性，并且仅存在于偶数维空间中； 正交矩阵和正交-Hermitian的 Ortho-Skew-Hermitian 复矩阵。Ortho-Skew和Ortho-Skew-Hermitian矩阵表示标量虚数到矩阵场的扩展。