On the determinant of quaternionic polynomial matrices and its application to system stability

The research reported in this paper is motivated by the study of stability for linear dynamical systems with quaternionic coefficients. These systems can be used to model several physical phenomena, for instance, in areas such as robotics and quantum mechanics. More concretely, quaternions are a powerful tool in the description of rotations [1]. There are situations, especially in robotics, where the rotation of a rigid body depends on time, and this dynamics is advantageously written in terms of quaternionic differential or difference equations. The effort to control the rotation dynamics motivates the study of these equations from a system theoretic point of view (see, for instance, [2]). Another motivation stems from quantum mechanics, where a quaternionic formulation of the Schrödinger equation has been proposed in the sixties along with experiments to check the existence of quaternionic potentials (see, for instance, [3]). This theory leads to differential equations with quaternionic coefficients [4].