On SVD and Polar Decomposition in Real and Complexified Clifford Algebras

Advances in Applied Clifford Algebras. 2024. Vol. 34. Article 23.

Широков Д. С.

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.

Ключевые слова: Singular Value Decomposition, SVD, symplectic group, Clifford algebra, orthogonal groups, geometric algebra, unitary group, polar decomposition