Research On The Iterative Methods For Two Kinds Of Quateranion Matrix Equations

Quaternion and quaternion matrix equations have important applications in quantum physics,color image restoration,kernel filtering,signal statistics and neural networks.At present,the research on the general solution and structural solution of quaternion matrix equation have become a hot topic in the direction of matrix algebra.In this paper,the numerical iterative problems of sub-positive definite matrix equation AX = B and quadratic system XAX-BX = P on the quaternion field are studied by using the special structure of quaternion matrix,matrix decomposition techniques and the eigenvalues of matrices and other related theories.The content of this article consists of the following 5 chapters:Chapter 1 mainly introduces the research background and research status of quaternion and quaternion matrix equations,proposes the main research content,and gives some basic concepts and lemmas.Chapter 2 discusses the numerical solution of the quaternion sub-positive definite matrix equation AX = B by using the split iterative method with parameters,and gives the convergence analysis of its iteration and the method of estimating quasi-optimal parameters.Chapter 3 studies the Modified Positive Definite and Skew-selfconjugate Splitting(MPSS)iteration of the quaternion sub-positive definite system AX = B whose coefficient matrix A is a strongly skew self-conjugate part,and gives the convergence analysis of the method.Chapter 4 discusses the existence and iterative algorithm of the positive definite selfconjugate solution of quadratic quaternion system XAX-BX = P and construct three convergent iterative schemes for different cases of coefficient matrix.At the same time,according to the characteristics of each iteration,the selection method of the initial matrix is given.Chapters 5 briefly summarize the main research work and point out the problems worthy of in-depth discussion.