On Circulant Matrix Solutions Of Quaternion Matrix Equations And Their Inverse Problems

The circulant matrices with special structure has been widely applied in manyfelds such as Signal Processing, image reconstruction, Coding Technology and so on.At present, complex circulant matrix has been completed results. But there are fewdiscussions about circulant matrix over quaternion division ring.On the thesis, the left and right eigenvalue and their inverse problems of quaternionmatrices are investigated, and circulant matrix solutions to some class of quaternionmatrix equation and their optimal approximation.The present thesis is organized asfollows:Firstly, the eigenvalues and their inverse problem of quaternion circulant matrix arediscussed. By using the n-th primitive unit roots over complex feld and decompositionof a quaternion matrix, the left and right eigenvalues of the n-by-n quaternion circulantmatrix are described. At the same time, the solvability conditions for the invertibleof quaternion circulant matrix are obtained, and the expression of the solution on lefteigenvalues inverse problem of quaternion circulant matrix is given.Secondly, this part is focused on the problem of the circulant matrix solution ofquaternion matrix equation XB=C and its optimal approximation. By using thestructural formulas of a circulant matrix and the Moore-Penrose generalized inverse,the existence conditions of circulant matrix solution and its general solution formulaare obtained, and the set of the least-square solution under constrain condition withcirculant matrix is got. Meanwhile, in the above set of the least-square solution, theoptimal approximation solution to given the quaternion circulant matrix is derived,extending the range to solve the constrained matrix equation.Thirdly, this part discusses the cyclic matrix solution and optimal approximationproblem of the Sylvester equation AX XB=C over quaternion feld. By using thereal representation of a quaternion matrix and the specifc structure of a cyclic matrixand transforming a constrained quaternion matrix equation into an unconstrained realmatrix equation through Kronecker product, thus the necessary and sufcient conditionfor the existence of a cyclic matrix solution and the general solution of the equation are obtained. Meanwhile, in the solution set, the optimal approximation solution which hasminimal Frobenius norm for given quaternion cyclic matrix is derived.Fourthly, his part is focused on the problem of the circulant matrix solution of themixed-type Lyapunov matrix equations AX+XA*+BXB=C and the solution inverseproblems’solution of the equations AX+XA*=C and AXA*-X=C. For given thequaternion circulant matrix X and positive defnite matrix C，by using the propertyof the circulant matrix and the complex transformation method, the expressions of thesolution A of these two classes of the equation are obtained.