# The Moore-Penrose inverses of split quaternions

## TL;DR

This paper introduces the Moore-Penrose inverse for split quaternions and applies it to solve specific linear equations, also characterizing similarity and consimilarity of split quaternions.

## Contribution

It extends the concept of Moore-Penrose inverse to split quaternions and provides solutions to linear equations involving them, along with conditions for similarity and consimilarity.

## Key findings

- Solutions to equations $axb=d$, $xa=bx$, $xa=bar{x}$ using Moore-Penrose inverse
- Necessary and sufficient conditions for similarity of split quaternions
- Necessary and sufficient conditions for consimilarity of split quaternions

## Abstract

In this paper, we find the roots of lightlike quaternions. By introducing the concept of the Moore-Penrose inverse in split quaternions, we solve the linear equations $axb=d$, $xa=bx$ and $xa=b\bar{x}$. Also we obtain necessary and sufficient conditions for two split quaternions to be similar or consimilar.