Study Of Solutions And Algorithms For Some Quaternion Matrix Equations

Quaternions and quaternion matrices have extensive applications in stability theory and cybernetics, they are important tools in studying quantum mechanics, while quaternion matrix equations are the basis of corresponding mathematical physics models in studying quaternionic quantum mechanics. Some quaternion matrix equations are difficult to solve because of noncommutation of quaternion multiplication, so it is a quite meaningful thing that study solutions of mathematical physics models in quaternionic quantum mechanics under noncommutation.In this dissertation, we mainly do some studies in algorithms for linear and nonlinear quaternion matrix equations, and present expressions for solutions of some quaternion matrix equations.As the theoretical basis of Chapter 3 and Chapter 4, a series of algebraic properties of quaternion matrices are introduced, then necessary and sufficient conditions for secondary diagonalization matrices and a corresponding example are given in Chapter 2.In Chaper 3, by means of quaternion generalized singular value decomposition of quaternion matrices, we obtain a general expression that quaternion matrix equation AX A~* = B has centro-symmetric solutions or anti-centro-symmetric solutions, in addition, an expression for the optimal approximation solution to a given matrix is derived.In Chaper 4, using methods of complex representation and theories of Jordan canonical forms of quternion matrices, combining properties of solutions of matrix equations, we give an algorithm for a class of nonlinear quaternion matrix equations and present general expressions for solutions of them.