【论文】矩阵对角化及其应用.pdfVIP

矩阵对角化及其应用 摘 要 矩阵的理论与方法贯穿于行列式、线性方程组、线性空间、线性变换、二次型 等各个方面，高等代数的许多问题都可以转化为相应的矩阵问题来处理. 可对角化 矩阵 （即能够与对角矩阵相似的矩阵）作为一类特殊的矩阵，在理论上和应用上都 有着十分重要的意义． 本论文首先介绍了矩阵及其运算的基本概念和结论以及矩阵的特征值与特征 向量的概念，然后对可对角化矩阵的条件（包括充分条件和充要条件）及方法进行 了归纳总结，并给出具体例题以详细说明每一种方法的步骤．论文的另一部分内容 是总结可对角化矩阵的应用，包括在矩阵计算中的应用 （求方阵的高次幂和开方以 及矩阵函数等）、利用特征值求行列式的值、在微分方程、向量空间、线性变换等 方面的应用．通过本文总结的方法能使读者更加深刻的理解可对角化矩阵的本质及 不同对角化方法的区别和联系，进一步培养学生的发散思维，加强学生的计算能力． 关键词：矩阵，矩阵的对角化，充要条件，应用 I 矩阵对角化及其应用 Abstract Theories and solutions about matrix could be applied throughout any aspects of the determinant, linear equations, linear space, linear transformation, and quadratic forms etc., Solutions for many of the questions of advanced algebra can be converted into the method of corresponding matrix to figure out. The diagonalizable matrix (that is similar to diagonal matrix) as a special sort of matrix, is significant in both theories and applications, This paper introduces basic concepts and conclusions of matrices, as well as the concepts of eigenvalues and eigenvectors. Concrete examples are given in details in order to justify the procedures and purposes of each kind of solution. The other part of this paper is to summarize the applications of the diagonalizable matrix, such as: the high-power special matrix, using eigenvalues to figure out determinant, using eigenvalues and eigenvectors to figure out the matrix in reverse, identifying the similarity between matrices, and applications for many other aspects like vector space, linear transformation and so on. This paper can help readers understand the essence of the diagonalizable matrix distinction and links to different solutions in