Study On Eigenvalue And Inverse Eigenvalue Problems Of Quaternion Matrices

With the development of science and technology, the applications ofquaternion matrices have infiltrate into the field of structural mechanics, spacetechnology and so on. Relevant calculation problems of quaternion matrices havealso attracted the attention of people. In the paper, eigenvalue and inverseeigenvalue problems of quaternion matrices are investigated and the present paperis organized as follows:Firstly, using the principal right eigenvalues of a quaternion matrix, theabsolute value of difference between any two right eigenvalues is described, andgive the definition of spread for a quaternion matrix. Then, by property ofcomplexification operator of the quaternion matrices, the estimates of upperbounds for the spread of a quaternion matrix are obtained. The numeral examplesshow the feasibility of the method.Secondly, we discuss convergent splitting of a sub-positive definite matrixover quaternion field. Meanwhile, the QSOR iterative algorithms of the sub-positive definite linear systems AX=B over quaternion field are structured, andconvergence of the algorithm is described by using right eigenvalue maximumnorm of the quaternion matrix, and the optimal range of parameters are obtained.Next, applying the structure preserving property of the complex representationoperation of the quaternion matrix, QSOR is transformed into the iteration overcomplex field, thus realizes numerical solutions of the linear systems.Thirdly, inverse problem solution and minimum norm and the least squaressolution of the UALE over quaternion field are investigated. The basic researchorder is: the necessary and sufficient conditions for the existence a solution of theinverse problem is presented using theory of matrix diagonalization, and theexpression of general solution is given. Next, in the set of general solution, byusing singular value decomposition of a matrix, its minimum Frobenius norm andthe least squares solution is obtained.Fourthly, we investigate the problems of the construction of arrow-likeself-conjugate quaternion matrix, and arrow-like self-conjugate positive definitequaternion matrix from two right eigenpairs. Necessary and sufficient conditions for the existence of a unique solution of these problems, as well as the expressionsof the solutions are derived. And then we investigate the problem of theconstruction of tridiagonal quaternion matrix from two right eigenpairs. Thenecessary and sufficient conditions for the existence of a solution of the problem,and the expressions of the solution are given. Moreover, the existence conditionsand calculation method for tridiagonal self-conjugate and tridiagonal positivedefinite quaternion matrix are also derived.