On Rank of Multivectors in Geometric Algebras

TL;DR

This paper introduces a novel way to determine the rank of multivectors in Clifford geometric algebras using geometric algebra operations, characteristic polynomials, and SVD, avoiding matrix representations.

Contribution

It presents a new method for calculating multivector rank directly within geometric algebra, expanding tools for applications in science and engineering.

Findings

Defined multivector rank in geometric algebra context

Developed rank computation method using characteristic polynomial and SVD

Applicable to various fields like computer science, engineering, and physics

Abstract

We introduce the notion of rank of multivector in Clifford geometric algebras of arbitrary dimension without using the corresponding matrix representations and using only geometric algebra operations. We use the concepts of characteristic polynomial in geometric algebras and the method of SVD. The results can be used in various applications of geometric algebras in computer science, engineering, and physics.