Sylvester矩阵方程相关问题研究

发布时间：2018-05-18 05:48

本文选题：Sylvester矩阵方程 + T-Sylvester矩阵方程　；参考：《哈尔滨工业大学》2017年博士论文

【摘要】：矩阵方程是一个重要的研究课题,在系统理论、自动控制、稳定性理论和优化理论等领域有着广泛的应用。本文将围绕Sylvester矩阵方程的相关问题进行研究。首先,研究了2可逆的交换环上的T-Sylvester矩阵方程的可解性问题。通过将问题转化到Artin环上,构造两个模并建立它们之间的映射来分析问题。推广了复数域上Wimmer给出的关于T-Sylvester矩阵方程有解的充要条件。还讨论了离散形式的T-Sylvester矩阵方程的可解性,给出了其存在解的充要条件。其次,研究了域上的一元多项式环F[λ]上的广义Sylvester矩阵方程的可解性以及解的性质。通过矩阵多项式的带余除法将问题转化为在低次的情形下讨论域上的一元多项式环上的广义Sylvester矩阵方程解的问题。利用矩阵张量积的相关技巧得到了多项式环上的广义Sylvester矩阵方程有解的充要条件,使得问题转化为未知参数的个数较少的新的方程。再次,研究了广义Lyapunov算子的最大奇异值所对应奇异向量的对称性。构造了反例,指出关于经典的Lyapunov算子最大奇异值所对应奇异向量是对称矩阵的现有证明是错误的,因此这仍是一个猜测。证明了在矩阵阶数小于6的时候该猜测是正确的。对于广义Lyapunov算子在矩阵的阶数小于4时,证明了其最大奇异值所对应奇异向量可以取为对称矩阵。而当矩阵的阶数大于或等于4时,给出例子说明这并不总是成立的。最后,研究了离散型Lyapunov算子和广义Lyapunov算子的最小奇异值所对应奇异向量的对称性。在矩阵的阶数不大于2的情形,广义Lyapunov算子的最小奇异值所对应奇异向量总可以取为对称矩阵。而当阶数大于2时,给出例子说明这个奇异向量并不总是可以取为对称矩阵。证明了,对于稳定的矩阵对,广义的Lyapunov算子的的最小奇异值所对应奇异向量总是可以取为对称矩阵。
[Abstract]:Matrix equation is an important research subject, which has been widely used in the fields of system theory, automatic control, stability theory and optimization theory. In this paper, the related problems of Sylvester matrix equation are studied. Firstly, the solvability of T-Sylvester matrix equations over 2 invertible commutative rings is studied. The problem is analyzed by transforming the problem to a Artin ring, constructing two modules and establishing a mapping between them. In this paper, the necessary and sufficient conditions for T-Sylvester matrix equation to have solutions given by Wimmer on complex field are generalized. The solvability of discrete T-Sylvester matrix equations is also discussed, and the necessary and sufficient conditions for its existence are given. Secondly, the solvability and the properties of the generalized Sylvester matrix equation over a polynomial ring F [位] are studied. The problem is transformed into the solution of the generalized Sylvester matrix equation on the univariate polynomial ring over the domain in the case of low degree by the method of codivision of matrix polynomials. By using the technique of matrix tensor product, the sufficient and necessary conditions for the generalized Sylvester matrix equation over polynomial rings to have solutions are obtained, so that the problem can be transformed into a new equation with fewer unknown parameters. Thirdly, the symmetry of the singular vector corresponding to the maximum singular value of the generalized Lyapunov operator is studied. In this paper, we construct a counterexample and point out that the existing proof that the singular vector corresponding to the maximum singular value of the classical Lyapunov operator is a symmetric matrix is wrong, so this is still a conjecture. It is proved that the conjecture is correct when the order of the matrix is less than 6. When the order of a generalized Lyapunov operator is less than 4, it is proved that the singular vector corresponding to its maximum singular value can be regarded as a symmetric matrix. When the order of a matrix is greater than or equal to 4, an example is given to show that this is not always true. Finally, the symmetry of singular vector corresponding to the minimum singular value of discrete Lyapunov operator and generalized Lyapunov operator is studied. When the order of a matrix is not greater than 2, the singular vector corresponding to the minimum singular value of the generalized Lyapunov operator can always be taken as a symmetric matrix. When the order is greater than 2, an example is given to show that the singular vector can not always be taken as a symmetric matrix. It is proved that the singular vector corresponding to the minimum singular value of the generalized Lyapunov operator can always be taken as a symmetric matrix for a stable pair of matrices.
【学位授予单位】：哈尔滨工业大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O241.6

【相似文献】