Canonic form of linear quaternion functions

TL;DR

This paper introduces a simple matrix-based method to derive the canonical form of linear quaternion functions, building on recent work that limits such functions to at most four quaternion coefficients.

Contribution

A new, straightforward numerical approach using matrices is proposed for obtaining the canonical form of linear quaternion functions, providing an alternative proof.

Findings

Method efficiently computes quaternion coefficients

Confirms the maximum of four coefficients for canonical form

Simplifies previous approaches to quaternion function representation

Abstract

The general linear quaternion function of degree one is a sum of terms with quaternion coefficients on the left and right. The paper considers the canonic form of such a function, and builds on the recent work of Todd Ell, who has shown that any such function may be represented using at most four quaternion coefficients. In this paper, a new and simple method is presented for obtaining these coefficients numerically using a matrix approach which also gives an alternative proof of the canonic forms.