利用循环矩阵的性质寻找矩阵对角化的方法外文翻译.doc

大连交通大学2014届本科生毕业设计（论文）外文翻译The cyclic nature of the matrix diagonalization method to find a matrix From:.hk/books?hl=zh-CN&id=7z5HPgAACAAJ Author：David C.Lay May 1990Diagonal matrix is a kind of simple matrix, which as quantum mechanics, radio, electronics and information engineering, computer plays an important role in many fields. Matrix diagonalization of great practical value. This paper describes the use of natures cycle looking matrix diagonalization matrix method.The basic definition of the text involvedDefinition 1: Let A is a square matrix of order n, if there is a nonzero number and n-dimensional vector , making is called an eigenvalue of the matrix A, is As a feature vector belongs to one of the.Definition 2:Let Ais the n-order square, known the determinantAs the characteristic polynomial of A, denoted , and = 0 is called A characteristic equation.Definition 3:The n-order square A is called reversible, if there are n-order matrix B, such that , where I is the identity matrix of order n.Definition 4: A, B is a matrix of order n, n-order reversible if there exists a matrix P, so that, then A and B are similar, called B of A similar matrix.Definition 5: If the number of domain P, on the existence of an n-level matrix A invertible matrix T ,make as diagonal matrix,we called matrix A over a number field P diagonalizable; When A keratosis when can we say that the A diagonalization, referring to seek reversible matrix T so for the angular matrix.Man involved in fundamental theoremTheorem 1:N order matrix A necessary and sufficient condition similar to a diagonal matrix A by n linearly independent eigenvectors, and when A is similar to a diagonal matrix whens main diagonal elements are all eigenvalues.Corollary 1: Matrix A necessary and sufficient condition similar to a diagonal matrix of eigenvalues belonging to each of linearly independent eigenvectors A number exactly equal to the multiplicity of the eigenvalues.Theorem 2: If the matrix A of order n has n mutually different eigenvalues (ie eigenvalues of A are single characteristic value), then A will be similar to a diagonal matrix.The cyclic nature of the matrix diagonalization method to find a matrix1.A basic similarity in through the back diagonal matrixn matrix P through the back called a basic array.it meets on the following properties: the basic matrix obtained through the back of the characteristic polynomial P is:Because the characteristic polynomial there are n distinct characteristic roots:So, basically through the back array P is similar to a diagonal matrix.The following feature vector obtained: takeThere（because）, sois characterized by the rootP corresponding to the eigenvectors.As a matrix:,Because As Determinant，So T reversible, then: .2. Through the back similar to the diagonal matrixMatrixreferred back to the front through.can find the basic matrix of polynomials out through the back:Let： , so back through the array can be diagonalized.3.Arbitrary n matrix A can keratosis is a necessary and sufficient condition A similar array of an n-order through the back .Proof:Adequacy: If A is similar to the array through the back so that the presence of reversible matrix C existbutso ie A similar diagonal.Necessity: If A can be diagonalized, which makes the presence of reversible square.With a polynomial of degree n-1As an equation as follows:,Namely: determinant factor in the equation is Determinant,thus the Law known by the unique solution of the equation Let of orderThe polynomial of degree n-1 is:,take matrix, Where P is the basic matrix through the back,thus Q is through the back front, and there is So, , that A similar to Q.array through the back.利用循环矩阵的性质寻找矩阵对角化的方法From:.hk/books?hl=zh-CN&id=7z5HPgAACAAJ 作者：David C.Lay 1990年5月对角矩阵是一类最简单的矩阵，它在许多领域如量子力学、无线电、电子信息工程、计算机等中起着重要的作用。研究矩阵对角化问题很有实用价值。本文主要介绍利用循环矩阵的性质寻找矩阵对角化的方法。文中涉及的基本定义定义1:设是阶方阵,如果存在数和维非零向量,使得则称是矩阵的一个特征值, 是的属于的一个特征向量。定义2:设为阶方阵,称行列式为的特征多项式,记为,而称为的特征方程。 定义3:阶方阵称为可逆的,如果存在阶方阵,使得,其中是阶单位矩阵。定义4:设,是阶方阵,若存在阶可逆矩阵,使得,则称与相似,称为的相似矩阵。 定义5:如果数域上,对级矩阵存在一个可逆矩阵使为对角形矩阵,则称矩阵在数域上可对角化;当可对角化时,我们说将对角化,即指求可逆矩阵使为对角形矩阵。文中涉及的基本定理定理1:阶方阵相似于对角矩阵的充分必要条件是由个线性无关的特征向量,且当相似于对角矩阵时,的主对角线元素就是的全部特征值。推论1:方阵相似于对角矩阵的充分必要条件是的属于每个特征值的线性无关的特征向量个数正好等于该特征值的重数。定理2: 如果阶方阵有个互不相同的特征值（即的特征值都是单特征值）,则必相似于对角矩阵。利用循环矩阵性质寻找矩阵对角化的方法1.基本循回阵相似于对角阵阶矩阵称为基本循回阵。它满足于如下性质： 求出基本循回阵的特征多项式：因为特征多项式有个不同特征根：所以,基本循回阵相似于对角阵。下面求出特征向量：取则有（因）, 从而为特征根对应的的特征向量.作矩阵：,因为为行列式,所以 可