On the existence of left and right eigenvalues

TL;DR

This paper proves the existence of left and right eigenvalues for matrices over certain finite-dimensional real algebras that contain complex numbers, extending classical eigenvalue results to non-commutative and alternative algebras.

Contribution

It establishes the existence of left and right eigenvalues for matrices over finite-dimensional real algebras containing complex numbers, including octonions and Cayley-Dickson algebras.

Findings

Eigenvalues exist for matrices over algebras with complex subalgebras.

Results apply to octonions and Cayley-Dickson algebras.

Provides a foundation for spectral theory in non-associative algebras.

Abstract

In this note, we consider arbitrary finite-dimensional real algebras containing a copy of complex numbers. It is proved that matrices with entries from an arbitrary finite-dimensional real algebra containing a square root of negative one in its left (resp. right) associate set have left (resp. right) eigenvalues. A quick consequence of our main result is the existence of left and right eigenvalues for matrices with entries from finite-dimensional alternatives algebras containing a copy of complex numbers, e.g., octonions, and more generally matrices with entries from the real Cayley-Dickson algebras.