The Moore-Penrose Inverses of Clifford Algebra $C\ell_{1,2}$

TL;DR

This paper establishes a ring isomorphism between the Clifford algebra $C ell_{1,2}$ and matrices, introduces the Moore-Penrose inverse in this algebra, and applies it to solve linear equations and analyze similarity of elements.

Contribution

It defines the Moore-Penrose inverse within $C ell_{1,2}$ using a matrix isomorphism, enabling new solutions for linear equations and similarity conditions.

Findings

Ring isomorphism between $C ell_{1,2}$ and matrix rings

Definition of Moore-Penrose inverse in $C ell_{1,2}$

Conditions for element similarity in $C ell_{1,2}$

Abstract

In this paper, we introduce a ring isomorphism between the Clifford algebra $C\ell_{1,2}$ and a ring of matrices. By such a ring isomorphism, we introduce the concept of the Moore-Penrose inverse in Clifford algebra $C\ell_{1,2}$. Using the Moore-Penrose inverse, we solve the linear equation $axb=d$ in $C\ell_{1,2}$. We also obtain necessary and sufficient conditions for two numbers in $C\ell_{1,2}$ to be similar.