Octonionic Quadratic Equations

TL;DR

This paper extends the explicit solution formulas for quadratic equations from quaternions to octonions, addressing the challenges posed by non-associativity and non-commutativity in octonionic algebra.

Contribution

It derives explicit formulas for solving monic left octonionic quadratic equations and applies these to determine the spectrum of octonionic matrices.

Findings

Explicit formulas for octonionic quadratic roots

Solution of monic left octonionic quadratic equations

Representation of octonionic matrix spectra

Abstract

There are four division algebras over $\mathbb{R}$, namely real numbers, complex numbers, quaternions, and octonions. Lack of commutativity and associativity make it difficult to investigate algebraic and geometric properties of octonions. It does not make sense to ask, for example, whether the equation $x^2+1=0$ is solvable, without specifying the field in which we want the solutions to be lie. The equation $x^2+1=0$ has no solutions in $\mathbb{R}$, which is to say, there are no real numbers satisfying this equation. On the other hand, there are complex numbers which do satisfy this equation in the field $\mathbb{C}$ of all complex numbers. How about if we extend the same idea to other two normed division algebras quaternions and octonions. Liping Huang and Wasin So derive explicit formulas for computing the roots of quaternionic quadratic equations. We extend their work to octonionic case and solve monic left octonionic quadratic equation of the form $x^2+bx+c=0$, where $a,b$ are octonions in general.[ We called this form of quadratic equation as left octonion quadratic equation because we can consider $x^2+xb+c=0$ as a different case due to non-commutativity of octonions]. Finally, we represent the left spectrum of $2\times2$ octonionic matrix as a set of solutions to a corresponding octonionic quadratic equation, which is an application of deriving explicit formulas for computing the roots of octonionic quadratic equations.