Iterative Algorithms Of Sylvester Matrix Equations And The Solutions To The Quaternion Matrix Equations

The contents of this thesis are divided into two parts:the first part is concerned with the iterative algorithm researches in real or complex filed, which are included in Chapter2and Chapter3:the second one is devoted to the solution to the quaternion matrix equation, see Chapter4and Chapter5for details. It mainly includes:1, Iterative algorithm to coupled Sylvester-transpose matrix equationsTwo iterative algorithms are proposed to investigate the iterative solutions to the coupled Sylvester-transpose matrix equation. One iterative algorithm is the hierarchical identification principle iterative algorithm. It is proved that the iterative solution consis-tently converges to the exact solution for any initial matrix group. Meanwhile, sufficient conditions are derived to guarantee that the iterative solutions given by the proposed algorithm converge to the exact solution for any initial matrix group and we extend the conclusion in the reference [134]. Next, a numerical example is given to illustrate the efficiency of the proposed approach. By extending the idea of conjugate gradient method, the other iterative algorithm, which is called finite iterative algorithm, is constructed. When the considered matrix equations is consistent, for any initial matrix group, a solu-tion group can be obtained within finite iteration steps in the absence of roundoff errors. The least Frobenius norm solution group of the coupled Sylvester-transpose matrix equa-tions can be derived when a suitable initial matrix groups is chosen. By applying the proposed algorithm, the optimal approximation solution group to a given matrix group can be obtained by finding the least Frobenius norm solution group of new general coupled matrix equations and we extend the conclusion in the reference [118]. Finally, a numerical example is given to show the effectiveness of the proposed algorithm.2. An efficient algorithm for solving extended Sylvester-conjugate trans-pose matrix equationsBy using the hierarchical identification principle, an iterative algorithm is presented for solving this class of extended Sylvester-conjugate transpose matrix equation (including the conjugate transpose matrix equation X+AXHB=F[142] as special case). It is proved that the iterative solution consistently converges to the exact solution for any initial matrix. Meanwhile, by means of a real representation of a complex matrix, suffcient conditions are derived to guarantee that the iterative solutions given by the proposed algorithm converge to the exact solution for any initial matrix. Finally, a numerical example is given to illustrate the efficiency of the proposed approach.3. On solutions to the quaternion matrix equationsBy means of Kronecker map and complex representation of a quaternion matrix, some explicit solutions to the quaternion matrix equations X F-AX=C and X-AXF=C are established. Two practical algorithms for these two equations are given. Meanwhile, the examples are proposed to illustrate the effectiveness of the proposed method. In addition, based on the Kronecker map and real representation of a quaternion matrix, the solution and the existence condition of the solution to the quaternion matrix equation AXB-CXD=E are given. Jamson's theorem of complex field is popularized to the quaternion field. Furthermore, a numerical example is given to show the effectiveness of the proposed algorithm.4. Iterative solution to coupled quaternion matrix equationsA real inner product in quaternion space is defined. A quaternion matrix norm which is a generalization of complex matrix norm is introduced. Based on the classical complex conjugate graduate iterative algorithm, an iterative algorithm is proposed. When the considered coupled quaternion matrix equations is consistent, it is proven by using a real inner product in quaternion space as a tool that a solution can be obtained within finite iteration steps for any initial quaternion matrices in the absence of round-off errors and the least Frobenius norm solution can be derived by choosing a special kind of initial quaternion matrices. Furthermore, the optimal approximation solution to a given quater-nion matrix can be derived. Finally, a numerical example is given to show the efficiency of the presented iterative method.