On left spectrum of a split quaternionic matrix

TL;DR

This paper investigates the left spectrum of 2x2 matrices over split quaternions, a noncommutative algebra, by deriving formulas for solving specific equations to better understand spectral properties.

Contribution

It introduces methods to analyze the left spectrum of split quaternion matrices, addressing a less explored area in noncommutative algebra.

Findings

Formulas for solving $ax=b$ over split quaternions

Formulas for solving quadratic equations over split quaternions

Initial understanding of the left spectrum for 2x2 split quaternion matrices

Abstract

In noncommutative and nondivision algebra, left spectrum of matrices are less known and is not easy to handle. Split quaternion algebra is a noncommutative and nondivision algebra. In this paper, by the formulas of solving the equations $ax=b$ and $ax^2+bx+c=0$ over split quaternions, we make an attempt to understand the left spectrum of a split quaternion matrix of order 2.