The Moore-Penrose Inverses of Clifford Algebra $C\ell_2$

TL;DR

This paper establishes a matrix-based representation of Clifford algebra $C ext{l}_2$, introduces the Moore-Penrose inverse within it, and solves related linear equations, providing new algebraic insights.

Contribution

It introduces the Moore-Penrose inverse in $C ext{l}_2$ via a ring isomorphism and solves key linear equations in this algebra.

Findings

Representation of $C ext{l}_2$ elements by real matrices

Conditions for similarity and pseudosimilarity in $C ext{l}_2$

Solutions to linear equations involving $C ext{l}_2$ elements

Abstract

In this paper, we introduce a ring isomorphism between the Clifford algebra $C\ell_2$ and a ring of matrices, and represent the elements in $C\ell_2$ by real matrices. By such a ring isomorphism, we introduce the concept of the Moore-Penrose inverse in Clifford algebra $C\ell_2$. we solve the linear equation $axb=d$, $ax=xb$ and $ax=\bar{x}b$. We also obtain necessary and sufficient conditions for two numbers in $C\ell_2$ to be similar and pseudosimilar.