General Polynomials and Eigenvalues Over Cayley--Dickson Algebras

Abstract

In this paper, we provide an explicit method for determining the zero set of monic quadratic general polynomials over Cayley--Dickson algebras with any base field of characteristic not equal to $2$, which reduces the problem to finding the roots of univariate polynomials over the base field. For the special case of locally-complex Cayley--Dickson algebras over the reals, we prove that every such polynomial has a root. We prove the existence of right eigenvalues for any $2\times 2$ matrix over Cayley's octonions (the real octonion division algebra), and our root-finding method allows the computation of some of the right eigenvalues.