Research On Several Kinds Of Constrained Matrix Equations Over The Equaternion Field

The problems of solving linear matrix equations and the corresponding least squares problems have been a hot topic in the field of numerical algebra in recent years, and have been widely applied in many fields such as structural design, system identification, structuraldynamics, automatic control theory, vibration theory. The least squares solutions of linear matrix equation are in general not unique, however, the least squares solution with the least norm is in general unique, where the norm is the Frobenius norm. In this dissertation, we systematically study the least squares solution of some kinds of quaternionmatrix equations with the least norm, respectively. They can be described in details as follows:ProblemⅠFind X∈S such thatwhere S is the least squares constraint solution set of quaternion matrix equationAXB = C.ProblemⅡFind [X, Y]∈S such thatwhere S is the least squares constraint solution set of quaternion matrix equationAXB + CYD = E.ProblemⅢFind [X, Y]∈S such thatwhere S is the least squares constraint solution set of quaternion matrix equationAXA~T + BYB~T = C.ProblemⅣFind X∈S such thatwhere S is the least squares constraint solution set of quaternion matrix equation(AXB,CXD) = (E,F). By using Moore-Penrose inverse, Kronecker product, vec operation and matrix decompositionmethods, we derive the solutions for ProblemⅠ,Ⅱ,ⅢandⅣ, respectively. The main works and creative points on this dissertation are as follows.1.The canonical correlation decomposition of quaternion matrices (CCD-Q) is established.Based on the projection theorem in the finite dimensional inner product space, by making use of the generalized singular value decomposition of quaternion matrices (GSVD-Q) and (CCD-Q) simultaneously, we transform the least squares problems of the abovementioned inconsistent quaternion matrix equations into the problems of solving the consistent matrix equations over given matrix set, and obtain the general expressions of the corresponding least squares solutions. Combining these expressions with the orthogonal invariance of the Frobenius norm of quaternion matrix, we obtain the expression of the solution of ProblemⅠ,Ⅱ,Ⅲ.2.By using Moore-Penrose inverse, Kronecker product, vec operation, the complex representations of quaternion matrices, and basic matrix of constrained matrix, the least squares constrained problems associated with quaternion matrix equations for Problem I, II, IV are changed into the least squares unconstrained problem associated with matrix equations, the expressions of corresponding least squares solutions, and the expressions of corresponding least squares solution with the least norm for ProblemⅠ,Ⅱ,Ⅳare derived.The conventional matrix decomposition methods have been used to find the least squares solutions of the abovementioned inconsistent matrix equations over given matrix set in many references, and the general expressions of these solutions were obtained. But it is difficult to determine the solution of ProblemⅠ,Ⅱ,Ⅲby utilizing these expressions due to the fact that the orthogonal invariance of Frobenius norm does not hold for the general nonsingular matrices. Based on the projection theorem in the finite dimensional inner product space and by making use of the generalized singular value decomposition of matrices (GSVD) and CCD simultaneously, a series of recent references overcome this difficulty skillfully, and obtain the expressions of the least square solution with the least norm abovementioned in ProblemsⅠ,ⅡandⅢ. We extend this skill into the skew field of quaternions. At first, we establish the canonical correlation decomposition of quaternion matrices (CCD-Q). Based on the projection theorem in the finite dimensional inner product space, by making use of the generalized singular value decomposition of quaternion matrices (GSVD-Q) and (CCD-Q) simultaneously, we transform the least squares problems of the abovementioned inconsistent quaternion matrix equations into the problems of solving the consistent matrix equations over given matrix set, and obtain the general expressions of the corresponding least squares solutions. Combining these expressions with the orthogonal invariance of the Frobenius norm of quaternion matrix, we obtain the expressions of the solutions of ProblemⅠ,Ⅱ,Ⅲ. This is the important improvement for the achievements of references.It seems that there are some difficulties in finding the least squares constrained (for example, symmetric) solutions of matrix equations AXB+CYD = E or (AXB, CXD) = (E, F) with the least norm by use of the conventional matrix decomposition methods. By using Moore-Penrose inverse, Kronecker product, vec operation, and basic matrix of constrained matrix, the least squares constrained problems for ProblemⅠ,Ⅱ,Ⅳare changed into the least squares unconstrained problems, the expressions of corresponding least squares solutions, and the expressions of corresponding least squares solution with the least norm for ProblemⅠ,Ⅱ,Ⅳare derived.