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arXiv · 2311.16704

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

Adam Chapman·Solomon Vishkautsan
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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.

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Adam Chapman, Solomon Vishkautsan. 2023-11-28. Roots and right factors of polynomials and left eigenvalues of matrices over Cayley-Dickson algebras. https://doi.org/10.46298/cm.12613

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