Linear equations in quaternionic variables

We study the quaternionic linear system which is composed out of terms of the form ln(x) := ∑n p=1 apxbp with quaternionic constants ap, bp and a variable number n of terms. In the first place we investigate one equation in one variable. If n = 2 the corresponding equation, which is normally called Sylvester’s equation will be treated completely by using only quaternionic algebra. For larger n a transition to the isomorphic (4 × 4) real matrix case is investigated. Sufficient conditions for non singularity will be obtained by using results from fixed point theorems. Connections to the Kronecker product are presented. The general case of a linear quaternionic system is treated, where each unknown is contained in a sum of the form mentioned above. As a tool the so-called column operator and its properties are used. An analogue of the Kronecker product for quaternionic systems involving terms of the form AXB is given.