Various Symmetric Solutions To Some Systems Of Linear Matrix Equations

In this dissertation, we investigate the general solutions, various symmetric solutions and the least squares solutions of some systems of matrix equations over regular ring, quaternion algebra and complex field. These results enrich and develop the matrix algebra.The dissertation is divided into 4 chapters. In Chapter 1, we introduce the research background and progresses of quaternion algebra and linear matrix equations, as well as the main work we have done in this dissertation. In addition, we introduce the preliminary knowledge used in this dissertation, such as some fundamental theory about regular ring and quaternion algebra, various symmetric matrices and the generalized inverses of partitioned matrices. In Chapter 2, by using the matrix techniques and matrix theory constructed in chapter 1, we investigate the matrix equations A₁X₁ =C₁, A₂X₁ = C₂,A₃X₂= C₃,A₄X₂=C₄,A₅X₁B₅ + A₆X₂B₆ = C₅ over regular ring, XB₁ = C₁,XB₂ = C₂,A₃XB₃ = C₃ and A₁X = C₁,XB₁ = C₂,A₂X = C₃,XB₂ = C₄,A₃XB₃ = C₅,A₄XB₄ = C₆ over quaternion algebra, respectively. We establish necessary and sufficient conditions for the existences of the solutions to these matrix equations. Furthermore, we give the expressions of the general solutions when the systems are consistent. These systems are not only meaningful in theory but also have important applicable values. In Chapter 3, applying the theory constructed in preceding two chapters, we consider the centrosymmetric(centroskew) solution to the matrix equations A_aX = C_a,A_bX = C_b,A_cXB_c = C_c over regular ring, the reflexive(antireflexive) solution to XB_a= C_a,A_bXB_b = C_b, P-symmetric solution to A_aX = C_a, A_bXB_b = C_B and the symmetric, persymmetric and centrosymmetric solutions to matrix equations A_aX = C_a, A_bX = C_b, A_cXB_c = C_c,A_aX = C_a, XB_b = C_b, A_cXB_c = C_c and XB_a = C_a, XB_b = C_b, A_cXB_c = C_c over quaternion algebra. We give the solvability conditions and the expressions of these various symmetric solutions to these systems mentioned above. In Chapter 4, we consider the (R, S)-conjugate solution, the optimal approximation problem and the least squares problems of a pair of matrix equations AX = B, XC = D over complex field. Moreover, we also consider the Procrustes problem of Hermitian R-conjugate matrix.