On Singular Value Decomposition and Polar Decomposition in Geometric Algebras

This paper is a brief note on the natural implementation of singular value decomposition (SVD) and polar decomposition of an arbitrary multivector in nondegenerate real (Clifford) geometric algebras of arbitrary dimension and signature. We naturally define these and other related structures (operation of Hermitian conjugation, Euclidean space, and Lie groups) in geometric algebras. The results can be used in various applications of geometric algebras in computer graphics, computer vision, data analysis, computer science, engineering, physics, big data, machine learning, etc.