On Rank of Multivectors in Geometric Algebras

TL;DR

This paper defines a matrix-free notion of rank for multivectors in complexified Clifford geometric algebras by developing a GA-based singular-value decomposition $M=U\Sigma V^\dagger$, where $U,V$ are GA-unitaries and $\Sigma$ lies in a fixed real subspace of dimension $N=2^{[\frac{n+1}{2}]}$. It also frames a determinant and characteristic-polynomial theory within GA via ${\rm Det}(M)=\det(\beta(M))$ and a GA-adapted Faddeev–LeVerrier recursion for coefficients $C_{(k)}$, tying rank to nonzero coefficients and to $M^\dagger M$. The authors prove that the GA rank ${\rm rank}(M)$ is independent of the chosen matrix representation and provide explicit rank criteria for small dimensions ($n=1,2,3,4$) using $M^\dagger M$ and the $C_{(k)}$. The results extend to real Clifford algebras through corresponding matrix realizations and open avenues for intrinsic GA-based analysis of rank, minors, and echelon-like structures in physics, engineering, and computer science applications.

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.

Theorems & Definitions (21)

• theorem 1: SVD in GA
• lemma 1
• proof
• theorem 2
• proof
• theorem 3
• proof
• lemma 2
• proof
• lemma 3