数值计算课程设计矩阵特征值和特征向量计算.docVIP

矩阵的特征值与特征向量的计算 摘 要 矩阵是高等代数学中的常见工具，也常见于统计分析等应用数学学科中.在物理学中，矩阵于电路学、力学、光学和量子物理中都有应用；计算机科学中，三维动画制作也需要用到矩阵. 矩阵的运算是数值分析领域的重要问题.将矩阵分解为简单矩阵的组合可以在理论和实际应用上简化矩阵的运算. 在本论文中，我们主要讨论矩阵的特征值和特征向量的计算，我们知道，有很多现实中的问题都可以用到矩阵特征值与特征向量计算的知识，比如，在物理、力学和工程技术方面有很多的应用，并且发挥着极其重要的作用.因为这些问题都可归结为求矩阵特征值的问题，具体到一些具体问题，如振动问题，物理中某些临界值的确定问题以及一些理论物理中的问题. 在本论文中，我们主要介绍求矩阵的特征值与特征向量的一些原理和方法，原理涉及高得代数中矩阵的相关定理，方法主要介绍冥法及反冥法，Jacobi方法和QR算法，并利用MATLAB，VC等软件编写相关算法的程序来求解相关问题，加以验证. 关键词： 矩阵；特征值；特征向量；冥法；反冥法；Jacobi方法；QR算法；VC软件；MATLAB软件 THE CALCULATIONS OF EIGENVALUE AND EIGENVECTOR OF MATRIX ABSTRACT The matrix is an usual tool in Advanced Algebra, which also used by applied mathematics such as Statistics Analysis. In Physics, we can see the important usage of matrix including Electric Circuits, Mechanics, Optics and Quantum Physics. Making three dimension needs matrix in Computer. The arithmetic of matrix is a very important part in Numerical Analysis. It can simplify the calculation of matrix that we decompose the matrix into several simple parts. In this thesis, we mainly talk about the calculation of eigenvalue and eigenvector of matrix. As we all know, there are lots of realistic problems which need the knowledge of the thesis to solve. We can see the important usage of matrix including Electric Circuits, Mechanics, Optics and Quantum Physics. It play an important role in these problems inferred above. Because these problems can regarded as the calculation of eigenvalue and eigenvector of matrix, like vibrating problems and critical value problems and so on. We primarily introduce the principle and approach of the calculation of eigenvalue and eigenvector of matrix that infer the relevant principle in Advanced Algebra. We mainly talk about iteration methods, Jacobi method and QR method by using MATLAB. Key words: Matrix；Eigenvalue；Eigenvector；Iteration methods; Jacobi method; QR method;MATLAB 目 录 1 引言............................................................1 2 相关定理。...........................................