四元数矩阵方程组特殊解理论研究与应用

Special solutions to system of matrix equations are not only important part of the matrix equation theory, but also plays an important role in the studying of comparison of linear normal experiment. By the existing methods, we can derive the Hermitian solutions, the nonnegative definite solutions and the least square solutions to system (1), but we can not get the real nonnegative definite solutions, as well as the unitary solutions to system (1).. As the finite generalized of complex matrix theory, quaternion matrix theory has became a new raising in matrix theory. In this problem, we will first generalize the inertia method to quaternion matrix and show the maximal and minimal inertias of some linear and nonlinear matrix expressions. Moreover, we will add some different constrict conditions to the variant matrix in the expression of the general solution, then the problem of searching the real nonnegative definite solutions to system (1) can be changed to finding some nonnegative definite solutions to a new system. It follows that the open problem can be solved. Noting that the unitary solutions and some special solutions to system (1) play important roles in linear model, we will show their applications in the comparison of linear experiment and linear growth model by rank and inertia methods.. This problem will not only improve the matrix equation theory, but also provide some powerful methods to the study of linear model.

矩阵方程组的特殊解理论不仅是矩阵方程理论的重要组成部分，而且在线性模型理论中起了重要作用。常规方法成功解决了矩阵方程组（1）的自共轭解、半正定解、最小二乘解等特殊解问题，但是无法得到其实半正定解、酉矩阵解等特殊解。.作为复数域矩阵方程组理论的有限维推广，四元数矩阵方程组理论成为当今矩阵方程理论研究热点。本项目，我们首先推广并应用四元数矩阵惯性指数方法，证明一些线性或非线性矩阵表达式的最大、最小惯性指数及其应用；然后采用逆向思维，并对方程组通解中自由矩阵加以限定，把原方程组实半正定解求解问题转换为新矩阵方程的半正定解问题，从而得到原方程组实半正定解。最后，注意到矩阵方程组的酉矩阵解等特殊解问题在线性模型理论中的应用，我们将利用矩阵的秩和惯性指数方法深入研究矩阵方程组及相关理论在正规实验比较等理论中的应用。.本项目的研究不仅能够完善矩阵方程理论，而且为线性模型理论研究提供强有力的理论基础。

项目研究计划中的内容基本都已完成，（1）针对矩阵方程理论：充分利用实半正定矩阵和半正定矩阵的关系，采用逆向思维，把方程组AX=C,XB=D的实半正定解得求解问题转换为其它矩阵方程组的半正定解问题，并且成功的解决矩阵方程组的公共半正定解和实部半正定解。我们利用秩方法给出了包含五个矩方程的矩阵方程组有Hermitian解的充要条件及通解表达式。此外，研究了Hilbert C*–modules中复杂的算子方程组理论理论，解决了复杂算子方程的通解和其它特殊解解问题。.（2）针对线性模型理论：我们充分运用矩阵运算技巧，证明一系列特殊的线性和非线性矩阵表达式的最大、最小秩和惯性指数，并且讨论了增长模型的最佳线性无偏估计、最小二乘估计和其它估计的优越性的判定条件.另外根据给出矩阵矩阵方程组有公共半正定解得充要条件和正定矩阵V 在某些特定假设检验条件下具有标准分布的充要条件；然后，把两个正规实验优越性的判定条件的验证，转换为矩阵惯性指数比较。