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

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.

References in corpus (5)

• The quaternionic Gauss-Lucas Theorem
• Roots and Dynamics of Octonion Polynomials
• Roots and right factors of polynomials and left eigenvalues of matrices over Cayley-Dickson algebras
• Von-Neumann finiteness and reversibility in some classes of non-associative algebras
• Roots and Critical Points of Polynomials over Cayley--Dickson Algebras

Cited by in corpus (2)

• Roots and right factors of polynomials and left eigenvalues of matrices over Cayley-Dickson algebras
• The algebraic and geometric classification of right alternative and semi-alternative algebras