Basis-free Solution to General Linear Quaternionic Equation

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 $\mathbf{i},\mathbf{j},\mathbf{k}$ 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.