In this paper, we present a natural implementation of singular value decomposition (SVD) and polar decomposition of an arbitrary multivector in nondegenerate real and complexified Clifford geometric algebras of arbitrary dimension and signature. The new theorems involve only operations in geometric algebras and do not involve matrix operations. We naturally define these and other related structures such as Hermitian conjugation, Euclidean space, and Lie groups in geometric algebras. The results can be used in various applications of geometric algebras in computer science, engineering, and physics.

中文翻译：

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实数和复数 Clifford 代数中的 SVD 和极分解

在本文中，我们提出了任意维度和签名的非简并实数和复数 Clifford 几何代数中任意多向量的奇异值分解（SVD）和极分解的自然实现。新定理只涉及几何代数中的运算，不涉及矩阵运算。我们自然地定义了这些以及其他相关的结构，例如几何代数中的埃尔米特共轭、欧几里得空间和李群。其结果可用于几何代数在计算机科学、工程和物理学中的各种应用。