On SVD and Polar Decomposition in Real and Complexified Clifford Algebras

IF 1.1 2区数学 Q2 MATHEMATICS, APPLIED

Advances in Applied Clifford Algebras Pub Date : 2024-05-27 DOI:10.1007/s00006-024-01328-9

Dmitry Shirokov

引用次数: 0

Abstract

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.

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论实数和复数克利福德代数中的 SVD 和极性分解

在本文中，我们提出了在任意维数和签名的非enerate实数和复数化克利福德几何代数中对任意多向量进行奇异值分解（SVD）和极性分解的自然实现方法。新定理只涉及几何代数的运算，不涉及矩阵运算。我们自然而然地定义了几何代数中的这些结构和其他相关结构，如赫尔墨斯共轭、欧氏空间和李群。这些结果可用于几何代数在计算机科学、工程学和物理学中的各种应用。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Advances in Applied Clifford Algebras 数学-物理：数学物理

CiteScore

2.20

自引率

13.30%

发文量

审稿时长

3 months

期刊介绍： Advances in Applied Clifford Algebras (AACA) publishes high-quality peer-reviewed research papers as well as expository and survey articles in the area of Clifford algebras and their applications to other branches of mathematics, physics, engineering, and related fields. The journal ensures rapid publication and is organized in six sections: Analysis, Differential Geometry and Dirac Operators, Mathematical Structures, Theoretical and Mathematical Physics, Applications, and Book Reviews.