Basis-free solution to general linear quaternionic equation

Abstract

ABSTRACTA linear quaternionic equation in one quaternionic variable is of the form , where the are given quaternionic coefficients. If introducing basis elements 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 , and found its solution 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.