An equivalent definition for multiplicities of a quaternionic eigenvalue

TL;DR

This paper introduces two new equivalent definitions for algebraic and geometric multiplicities of quaternionic eigenvalues that simplify proofs by avoiding root subspaces and Jordan forms.

Contribution

It presents alternative definitions for quaternionic eigenvalue multiplicities that are easier to work with and prove properties without relying on classical concepts.

Findings

Definitions are equivalent to classical ones.

Properties can be proved more easily.

No need for root subspaces or Jordan forms.

Abstract

In this paper we introduce two definitions for algebraic and geometric multiplicities of a quaternion right eigenvalue. This definitions are equivalent to the classical ones. However, differently from the usual definitions, the notions of root subspace and Jordan form of a quaternion matrix are not required and all the properties can be proved easily.