The Theoretical Framework Of Homogeneous Linear Quaternion Differential Equations

Quaternion was discovered by Sir William Rowan Hamilton, an Ireland mathemati-cian in 1843. Quaternion-valued differential equations (QDEs) have many applications in quantum mechanics, fluid mechanics, Frenet frame in differential geometry, kinematic modelling, attitude dynamics, Kalman filter design and spatial rigid body dynamics etc. However, the non-commutativity of the quaternion hinders the development of QDEs. Therefore, it is of great significance to study n-dimensional homogeneous linear QDEs.The thesis investigates the general theory of n-dimensional homogeneous linear QDEs, and obtains a method for computing fundamental matrix exp At of n-dimensional homogeneous linear QDEs with constant coefficients. Due to the non-commutativity of the quaternion, the definition of Caley determinant is no longer suitable for studying n-dimensional homogeneous linear QDEs. So we introduce a kind of definition of de-terminant based on symmetric group over the quaternion field. And a new definition of Wronskian is introduced based on double determinant. Due to the use of this defi-nition, the computation of the determinant is different. Hence we get a new Liouville formula whose proof is more complicate. We also get the conclusion that the set of all the solutions to the n-dimensional homogeneous linear QDEs is a n-dimensional right H-module (free module). Then a method for computing fundamental matrix exp At for any quaternion-valued coefficient matrix A is discussed. And finally, two examples are given to show the feasibility of the obtained algorithm.