Roots and right factors of polynomials and left eigenvalues of matrices over Cayley-Dickson algebras

TL;DR

This paper investigates roots and right factors of polynomials over Cayley-Dickson algebras, explores their relation to companion polynomials, and computes left eigenvalues of 2x2 octonion matrices, revealing specific cases where classical properties hold.

Contribution

It extends the understanding of polynomial roots and factorization over Cayley-Dickson algebras and provides explicit eigenvalue computations for octonion matrices.

Findings

Roots correspond to factors for linear and monic quadratic polynomials.

The root-factor relationship generally fails for higher-degree polynomials.

Explicit calculation of left eigenvalues for 2x2 octonion matrices.

Abstract

Over a composition algebra $A$, a polynomial $f(x) \in A[x]$ has a root $\alpha$ if and only $f(x)=g(x)\cdot (x-\alpha)$ for some $g(x) \in A[x]$. We examine whether this is true for general Cayley-Dickson algebras. The conclusion is that it is when $f(x)$ is linear or monic quadratic, but it is false in general. Similar questions about the connections between $f$ and its companion $C_f(x)=f(x)\cdot \overline{f(x)}$ are studied. Finally, we compute the left eigenvalues of $2\times 2$ octonion matrices.