On the existence of left and right eigenvalues
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.
Cite
@article{arxiv.2203.15919,
title = {On the existence of left and right eigenvalues},
author = {Bamdad R. Yahaghi},
journal= {arXiv preprint arXiv:2203.15919},
year = {2022}
}
Comments
Unfortunately, the main result of the note, namely Theorem 2.1 is false. Any scalar matrix with a quaternion or octonion that is not a complex number on its main diagonal cannot have left eigenvalues in complex numbers. At this point, I would like to thank Professor Alberto Elduque and an anonymous reviewer of the note for pointing out diagonal matrix counterexamples to me