# The Jordan-Chevalley decomposition and Jordan canonical form of a quaternionic linear operator

## TL;DR

This paper extends fundamental linear algebra concepts like Jordan-Chevalley decomposition and Jordan canonical form to quaternionic linear operators, providing new proofs and insights.

## Contribution

It introduces quaternionic analogues of classical decompositions and offers a more intrinsic, computationally efficient proof of the Jordan canonical form for quaternionic operators.

## Key findings

- Established additive and multiplicative Jordan-Chevalley decompositions for quaternionic operators
- Provided a new, intrinsic proof of the Jordan canonical form in the quaternionic setting
- Linked the decompositions via the exponential map

## Abstract

We introduce some basic notions and results for quaternionic linear operators analogous to those for complex linear operators. Our main result is to prove the additive and multiplicative Jordan-Chevalley decompositions for quaternionic linear operators, which are related by the exponential map. We also give a new proof of the theorem of Jordan canonical form for quaternionic linear operators, which is intrinsic and takes less computations than the known proofs for square quaternionic matrices.