矩阵Frobenius标准形的初等变换解法及其应用

矩阵的初等变换在求矩阵的秩、求方阵的逆矩阵、化实对称阵合同于对角阵、求解线性方程组等中均有重要的应用,本文给出了初等变换使方阵相似于Frobenius标准形的方法;该方法运算简单,容易实现,并为求方阵的特征多项式,化方阵为Jordan标准形及求出相应的相似变换阵带来极大的方便。     

分块矩阵的初等变换

本文将矩阵的初等变换的概念推广到分块矩阵上并建立了计算分块矩阵的逆矩阵和分块方阵的行列式的若干简易方法．     

分块矩阵的初等变换

本将矩阵的初等变换的概念推广到分块矩阵上并建立了计算分块矩阵的逆矩阵和分块方阵的行列式的若干简易方法。     

实对称矩阵对角化中正交矩阵的初等变换求法

对实对称矩阵正交对角化过程中正交矩阵的求解方法进行了研究,给出了利用初等变换求解正交矩阵的方法,该方法不需要通过特征方程求解特征值与特征向量,仅仅使用初等变换和Schmidt正交化方法.     

善用初等变换展开线性代数理论教学

初等变换是线性代数的基本变换,在线性代数课程中常常被用来计算,例如求解线性方程组、计算方阵的行列式、矩阵的求逆以及更一般的矩阵方程AX=B的求解、计算整数矩阵和域上多项式矩阵的Smith标准形、以及计算对称阵的相合标准形等.本文说明如何灵活利用初等变换,给出线性代数课程中一些重要理论结果的系统而又简洁的证明.     

矩阵的特征值和特征向量同时求解的一种方法

本文给出了通过λ－矩阵的初等变换，同时求得特征值和特征向量的一种方法。     

利用矩阵的初等变换求方阵的特征值

高阶方阵的特征多项式以及特征值的求得,在计算上往往有一定的难度.本文首先从理论上分析了存在一个上三角矩阵或者下三角矩阵与一个方阵相似;接着,提出了相似变换的概念,分析了相似变换中初等矩阵的选择方法;然后指出了利用相似变换在求方阵的特征多项式以及特征值时的方法,并列举若干实例给予了说明.     

矩阵初等变换及其逆阵

本文讨论的矩阵A为数域P上的可逆方阵,对A作初等变换: (i)对调i,j两行(列),这相当于用初等方阵     

关于方阵多项式的特征问题

对方阵及其矩阵多项式,给出了它们特征值、特征向量之间关系的刻画.     

正交变换及其反问题

一、引言在数值分析中,矩阵变换起着重要作用,特别在求解线性方程组的直接法以及特征值计算的方法中,矩阵变换是基本的。不同特点的变换矩阵将构造出不同特点的数值方法。如稳定的初等变换,Givens 变换(即平面旋转变换)以及 Householder 变换等,其中后两种是正交变换。设 R~(n×n)为所有 n 阶实方阵的全体,R~n 为 n 维稚实欧氏空间,||·||表示欧氏模,S 为     

Construction of Matrices with Prescribed Singular Values and Eigenvalues

Two issues concerning the construction of square matrices with prescribe singular values an eigenvalues are addressed. First, a necessary and sufficient condition for the existence of an n × n complex matrix with n given nonnegative numbers as singular values an m ( n) given complex numbers to be m of the eigenvalues is determined. This extends the classical result of Weyl and Horn treating the case when m = n. Second, an algorithm is given to generate a triangular matrix with prescribe singular values an eigenvalues. Unlike earlier algorithms, the eigenvalues can be arranged in any prescribe order on the diagonal. A slight modification of this algorithm allows one to construct a real matrix with specified real an complex conjugate eigenvalues an specified singular values. The construction is done by multiplication by diagonal unitary matrices, permutation matrices and rotation matrices. It is numerically stable and may be useful in developing test software for numerical linear algebra packages.     

关于稳定矩阵分解定理的一个简单证明

一个矩阵称为稳定的,如果这个矩阵的特征值全包含在单位开圆盘内.利用Parker关于复方阵的分解定理给出了稳定矩阵分解定理的一个简单证明,并对奇异值全部严格小于1的矩阵给出了类似的结论.     

Estimates for the Eigenvalues of Matrices

For matrices whose eigenvalues are real (such as Hermitian or real symmetric matrices), we derive simple explicit estimates for the maximal (λ_max) and the minimal (λ_min) eigenvalues in terms of determinants of order less than 3. For 3 × 3 matrices, we derive sharper estimates, which use det A but do not require to solve cubic equations.     

紧致交换矩阵半群

通过将矩阵同时对角化或同时上三角化的方法,给出有关紧致Abel矩阵半群以及紧致Hermite矩阵半群中矩阵的特征值的一些很好的刻画,证明了由可逆的Hermite矩阵构成的紧致矩阵半群中每个矩阵的特征值都是±1,Hermite矩阵单半群相似于对角矩阵半群,紧致交换矩阵半群的谱半径不超过1,等等.     

Some Spectral Properties of Metapositive Definite Matrice

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The Worst-Case GMRES for Normal Matrices

We study the convergence of GMRES for linear algebraic systems with normal matrices. In particular, we explore the standard bound based on a min-max approximation problem on the discrete set of the matrix eigenvalues. This bound is sharp, i.e. it is attainable by the GMRES residual norm. The question is how to evaluate or estimate the standard bound, and if it is possible to characterize the GMRES-related quantities for which this bound is attained (worst-case GMRES). In this paper we completely characterize the worst-case GMRES-related quantities in the next-to-last iteration step and evaluate the standard bound in terms of explicit polynomials involving the matrix eigenvalues. For a general iteration step, we develop a computable lower and upper bound on the standard bound. Our bounds allow us to study the worst-case GMRES residual norm as a function of the eigenvalue distribution. For hermitian matrices the lower bound is equal to the worst-case residual norm. In addition, numerical experiments show that the lower bound is generally very tight, and support our conjecture that it is to within a factor of 4/π of the actual worst-case residual norm. Since the worst-case residual norm in each step is to within a factor of the square root of the matrix size to what is considered an “average” residual norm, our results are of relevance beyond the worst case. This revised version was published online in July 2006 with corrections to the Cover Date.     

Matrices with prescribed Ritz values

On the way to establishing a commutative analog to the Gelfand-Kirillov theorem in Lie theory, Kostant and Wallach produced a decomposition of M(n) which we will describe in the language of linear algebra. The “Ritz values” of a matrix are the eigenvalues of its leading principal submatrices of order m=1,2,…,n. There is a unique unit upper Hessenberg matrix H with those eigenvalues. For real symmetric matrices with interlacing Ritz values, we extend their analysis to allow eigenvalues at successive levels to be equal. We also decide whether given Ritz values can come from a tridiagonal matrix.     

Distribution of Eigenvalues of Real Symmetric Palindromic Toeplitz Matrices and Circulant Matrices

Consider the ensemble of real symmetric Toeplitz matrices, each independent entry an i.i.d. random variable chosen from a fixed probability distribution p of mean 0, variance 1, and finite higher moments. Previous investigations showed that the limiting spectral measure (the density of normalized eigenvalues) converges weakly and almost surely, independent of p, to a distribution which is almost the standard Gaussian. The deviations from Gaussian behavior can be interpreted as arising from obstructions to solutions of Diophantine equations. We show that these obstructions vanish if instead one considers real symmetric palindromic Toeplitz matrices, matrices where the first row is a palindrome. A similar result was previously proved for a related circulant ensemble through an analysis of the explicit formulas for eigenvalues. By Cauchy’s interlacing property and the rank inequality, this ensemble has the same limiting spectral distribution as the palindromic Toeplitz matrices; a consequence of combining the two approaches is a version of the almost sure Central Limit Theorem. Thus our analysis of these Diophantine equations provides an alternate technique for proving limiting spectral measures for certain ensembles of circulant matrices. A. Massey’s current address: Department of Mathematics, UCLA, Los Angeles, CA 90095, USA. e-mail: amassey3102@math.ucla.edu.     

The First Order Asymptotics of the Extreme Eigenvectors of Certain Hermitian Toeplitz Matrices

The paper is concerned with Hermitian Toeplitz matrices generated by a class of unbounded symbols that emerge in several applications. The main result gives the third order asymptotics of the extreme eigenvalues and the first order asymptotics of the extreme eigenvectors of the matrices as their dimension increases to infinity. This work was partially supported by CONACYT projects 60160 and 80504, Mexico.