On solutions of singular Sylvester equations in quaternions
Hristina Radak, Christian Scheunert, Frank H. P. Fitzek
TL;DR
This work addresses solving singular Sylvester equations in the quaternionic setting, focusing on equations of the form $a x - x b = c$ where the operator is singular. The authors establish solvability conditions based on similarity of $a$ and $b$ and derive complete closed-form solutions for both the homogeneous ($c=0$) and inhomogeneous ($c\neq0$) cases, exploiting quaternion square roots to express nontrivial solution families. They show that nonzero solutions to the homogeneous singular equation exist precisely when $a \\sim b$, and provide explicit parametrizations that reduce to pure-quaternion forms; for the inhomogeneous case they give solvability criteria (including $a c = c b^*$ in suitable cases) and corresponding general solutions with explicit dependence on quaternion square roots. The results connect the quaternionic Sylvester problem to quaternion root theory, offering analytic, closed-form tools that can extend to related problems in quaternion analysis.
Abstract
The quaternionic equations ax-xb=0 and ax-xb=c are investigated, which are called homogeneous and inhomogeneous Sylvester equations, respectively. Conditions for the existence of solutions are provided. In addition, the general and nonzero solutions to these equations are derived applying quaternion square roots.