Basis-free Solution to General Linear Quaternionic Equation

TL;DR

This paper develops a general method to find basis-free solutions to linear quaternionic equations with any number of terms, extending previous work by Sylvester and Schwartz.

Contribution

It provides a unified solution approach for arbitrary-term quaternionic equations in the non-degenerate case, advancing prior specific solutions.

Findings

Explicit basis-free solutions for arbitrary-term equations

Extension of Sylvester's and Schwartz's methods

Solution applicable to non-degenerate cases

Abstract

A linear quaternionic equation in one quaternionic variable q is of the form $a_1 q b_1+a_2 q b_2+ ... +a_m q b_m = c$, where the $a_i, b_j, c$ are given quaternionic coefficients. If introducing basis elements $\bf i, j, k$ of pure quaternions, then the quaternionic equation becomes four linear equations in four unknowns over the reals, and solving such equations is trivial. On the other hand, finding a quaternionic rational function expression of the solution that involves only the input quaternionic coefficients and their conjugates, called a basis-free solution, is non-trivial. In 1884, Sylvester initiated the study of basis-free solution to linear quaternionic equation. He considered the three-termed equation $aq+qb=c$, and found its solution $q=(a^2+b\overline{b}+a(b+\overline{b}))^{-1}(ac+c\overline{b})$ by successive left and right multiplications. In 2013, Schwartz extended the technique to the four-termed equation, and obtained the basis-free solution in explicit form. This paper solves the general problem for arbitrary number of terms in the non-degenerate case.