毕业设计（论文）-矩阵方程AX+XB=D的极小范数最小二乘解精选.doc

题目：矩阵方程AX+XB=D的极小范数最小二乘解 姓 名 学 号 指导教师 专 业 学 院 摘 要解的研究与探索更是没有间断过，可见该方程的求解问题确实是一个非常重要的课题。 本文将围绕该命题展开讨论，利用矩阵的直积（Kronecker积）、按行拉直和矩阵的满秩分解等算法解出矩阵方程的极小范数最小二乘解，并且它可由Moore-Penrose逆表出。 关键词：矩阵方程 极小范数最小二乘解 矩阵的直积（Kroneck积） 满秩分解算 法 Moore-Penrose逆 ABSTRACT Matrix theory is the foundation of learning classical mathematics, as well as one of the most meaningful mathematical theories. It is not only an important branch of mathematics, but also has become a powerful tool of handling many relationships between the finite dimension space structures and quantities in varity fields of modern technology. The extensive use of computer has brought a bright prospect to the application of matrix theory. Many problem have close relation with matrix theory, such as system engineering, optimization method, stability theory, and so on. Nowadays, the research on matrix equations has been turning into one of the hottest topics in matrix theory . It is widely used in different areas, for example, biology, electrics, spectroscopy, vibration theory, linear optimal control, etc. And people have researched and explored the solution of matrix equation all the time, which means the subject study is absolutely crucial. This paper discusses the available solutions of matrix equation, using Kronecker product, full rank decomposition of matrix to obtain the minimal norm least squares solution of matrix equation . And it can be expressed by Moore-Penrose inverse. Key words: matrix equation; the minimum norm least squares solution; Kronecker product; full rank factorization algorithm; Moore-Penrose inverse 目 录 摘 要 1 ABSTRACT 2 1 绪论 4 1.1李雅普诺夫矩阵方程应用背景及研究现状 5 1.2 本文的主要工作 5 1.3 符号说明 6 2 预备知识 7 2.1 矩阵的范数 7 2.1.1 矩阵范数的定义 8 2.1.2 矩阵范数的性质 8 2.2 矩阵的直积及其应用 10 2.2.1 直积的概念 10 2.2.2 矩阵直积的性质 11 2.2.3 线性矩阵方程的可解性 11 2.3 矩阵的满秩分解 12 2.3.1 矩阵满秩分解的定义 12 2.3.2 矩阵满秩分解的性质 12 2.4 广义逆矩阵