数值方法课程设计幂法反幂法计算矩阵特征值和特征向量-附Matlab程序.docx

矩阵的特征值与特征向量计算 PAGE 矩阵的特征值与特征向量的计算 摘 要 物理，力学，工程技术中的很多问题在数学上都归结于求矩阵特征值的问题，例如振动问题（桥梁的振动，机械的振动，电磁振动等）、物理学中某些临界值的确定问题以及理论物理中的一些问题。矩阵特征值的计算在矩阵计算中是一个很重要的部分，本文使用幂法和反幂法分别求矩阵的按模最大，按模最小特征向量及对应的特征值。 幂法是一种计算矩阵主特征值的一种迭代法，它最大的优点是方法简单，对于稀疏矩阵比较合适，但有时收敛速度很慢。其基本思想是任取一个非零的初始向量。由所求矩阵构造一向量序列。再通过所构造的向量序列求出特征值和特征向量。 反幂法用来计算矩阵按模最小特征向量及其特征值，及计算对应于一个给定近似特征值的特征向量。本文中主要使用反幂法计算一个矩阵的按模最小特征向量及其对应的特征值。计算矩阵按模最小特征向量的基本思想是将其转化为求逆矩阵的按模最大特征向量。然后通过这个按模最大的特征向量反推出原矩阵的按模最小特征向量。 关键词： 矩阵；特征值；特征向量；冥法；反冥法 THE CALCULATIONS OF EIGENVALUE AND EIGENVECTOR OF MATRIX ABSTRACT Physics, mechanics, engineering technology in a lot of problems in mathematics are attributed to matrix eigenvalue problem, such as vibration (vibration of the bridge, mechanical vibration, electromagnetic vibration, etc.) in physics, some critical values determine problems and theoretical physics in some of the problems. Matrix eigenvalue calculation is a very important part in matrix computation. In this paper, we use the power method and inverse power method to calculate the maximum of the matrix, according to the minimum characteristic vector and the corresponding characteristic value. Power method is an iterative method to calculate the eigenvalues of a matrix. It has the advantage that the method is simple and suitable for sparse matrices, but sometimes the convergence rate is very slow. The basic idea is to take a non - zero initial vector. Construct a vector sequence from the matrix of the matrix. Then the eigenvalues and eigenvectors are obtained by using the constructed vector sequence. The inverse power method is used to calculate the minimum feature vectors and their eigenvalues of the matrix, and to calculate the eigenvalues of the matrix. In this paper, we use the inverse power method to calculate the minimum eigenvalue of a matrix and its corresponding eigenvalues. The basic idea of calculating the minimum characteristic vector of a matrix is to transform it to the maximum characteristi