二阶矩阵的特征值和特征向量常考题型例析

矩阵的特征值和特征向量是矩阵与变换的一个非常重要的内容,利用矩阵的特征值和特征向量,可以方便地计算多次矩阵变换的结果,而且在实际工程计算和工程控制中也发挥着重要作用.二阶矩阵的特征值和特征向量有两个基本内容.一是二阶矩阵的特征值和特征向量的概念：设A是一个二阶矩阵,如果对于实数λ,存在一个非零向量α,使得Aα=λα,那么λ称为A的一个特征值,而α称为A的属于特征值λ的一个特征向量.     

一维不定参数结构系统振动特征问题的摄动传递矩阵法

基于Riccati传递矩阵法，给出了一维不确定参数结构系统振动特征问题的二阶摄动计算方法，该方法适用于一般的一维结构系统的实数和复数特征问题的分析，并给出了结构振动特征的灵敏度计算公式．算例对转子的陀螺特征值问题进行了摄动分析，摄动结果和精确计算结果吻合良好．     

多参数二次特征值问题重特征值的灵敏度分析

本文研究解析依赖于多参数的二次特征值问题重特征值的灵敏度分析,得到了重特征值的方向导数,证明了相应的特征向量矩阵和特征值平均值的解析性,给出了其一阶偏导数的表达式．然后以这些结论为基础,定义了二次特征值问题重特征值及其不变子空间的灵敏度,并给出了确定二次特征值问题所含矩阵中敏感元素的方法．     

一类三对角矩阵的特征值和特征向量的研究

讨论了一种三对角矩阵的特征值和特征向量.按矩阵右下角对角元素的参数分为两类,得出特征值和特征向量的结论或数值算法.举例说明了算法的有效性.     

关于完全完备分配格上矩阵相对于特征值的特征向量

借助于伪补和矩阵的幂序列研究了完全完备分配格上矩阵相对于特征值的特征向量的计算方法,利用特征向量的性质证明了最大特征向量的计算公式,并给出了一般特征向量的计算方法.     

粘性阻尼线性振动系统的复模态特征值问题的一种新的矩阵摄动分析解法

本文提出了对粘性阻尼线性振动系统的复模态二次广义特征值问题进行高效近似求解的一种新的矩阵摄动分析方法,即先将阻尼矩阵分解为比例阻尼部分和非比例阻尼部分之和,并求得系统的比例阻尼实模态特征解;然后以此为初始值,将阻尼矩阵的非比例部分作为对其比例部分的小量修改,利用摄动分析方法简捷地得到系统的复模态特征值问题的近似解.这一新方法适用于振系阻尼分布不十分偏离比例阻尼情况的问题,因此对大阻尼(非过阻尼)振动系统也有效.这是它优于以前提出的基于无阻尼实模态特征解的类似摄动分析方法的重要特点.文中建立了复模态特征值和特征向量的二阶摄动解式,并通过算例证实了其有效性.此外还讨论了利用比例阻尼假定估计阻尼系统固有振动的复特征值的可行性.     

二次特征值问题特征对的一阶与二阶偏导数

本文研究了解析依赖于多参数的二次特征值问题特征对偏导数的计算.利用计算广义特征值问题特征向量偏导数的模态法.提出了一种计算二次特征值问题特征对一阶、二阶偏导数的方法.本文最后以弹簧质点阻尼系统为例验证了所给结论的正确性和方法的有效性.     

关于矩阵的特征值与特征向量同步求解问题

通过对矩阵进行行列互逆变换,同步求出矩阵特征值及特征向量,解决了不带参数求特征值问题,并给出一些新定理.     

Jordan标准形过渡矩阵的一种算法

给出Jordan定理的一个证明,以及Jordan标准形过渡矩阵的一种算法:求出一线性方程组解空间的基,解空间即是矩阵关于某特征值的特征向量、广义特征向量所张成的子空间,在该解空间中依次找出各特征向量及所对应的广义特征向量.一个8阶矩阵的计算实例表明算法简便实用.     

非对称阻尼系统特征对一阶导数与二阶导数的计算

提出了一种计算非对称阻尼系统特征对一阶、二阶导数的方法.该方法利用阻尼系统的特征向量计算特征对的导数,避免了状态空间中特征向量的使用,节省了计算量,且不要求系统所有特征值的互异性.最后以两个非对称阻尼系统进行数值试验,数值结果表明提出的方法是有效的.     

Rank-one modification of the symmetric eigenproblem

Summary An algorithm is presented for computing the eigensystem of the rank-one modification of a symmetric matrix with known eigensystem. The explicit computation of the updated eigenvectors and the treatment of multiple eigenvalues are discussed. The sensitivity of the computed eigenvectors to errors in the updated eigenvalues is shown by a perturbation analysis.Support for this research was provided by NSF grants MCS 75-06510 and MCS 76-03139Support for this research was provided by the Applied Mathematics Division, Argonne National Laboratory, Argonne, IL 60439, USA     

Hyman’s method revisited

The QR algorithm is considered one of the most reliable methods for computing matrix eigenpairs. However, it is unable to detect multiple eigenvalues and Jordan blocks. Matlab’s eigensolver returns heavily perturbed eigenvalues and eigenvectors in such cases and there is no hint for possible principal vectors. This paper calls attention to Hyman’s method as it is applicable for computing principal vectors and higher derivatives of the characteristic polynomial that may help to estimate multiplicity, an important information for more reliable computation. We suggest a test matrix collection for Jordan blocks. The first numerical tests with these matrices reveal that the computational problems are deeper than expected at the beginning of this work.     

Computation of multiple eigenvalues and generalized eigenvectors for matrices dependent on parameters

The paper develops Newton's method of finding multiple eigenvalues with one Jordan block and corresponding generalized eigenvectors for matrices dependent on parameters. It computes the nearest value of a parameter vector with a matrix having a multiple eigenvalue of given multiplicity. The method also works in the whole matrix space (in the absence of parameters). The approach is based on the versal deformation theory for matrices. Numerical examples are given. Copyright © 2005 John Wiley & Sons, Ltd.     

A block Newton method for nonlinear eigenvalue problems

We consider matrix eigenvalue problems that are nonlinear in the eigenvalue parameter. One of the most fundamental differences from the linear case is that distinct eigenvalues may have linearly dependent eigenvectors or even share the same eigenvector. This has been a severe hindrance in the development of general numerical schemes for computing several eigenvalues of a nonlinear eigenvalue problem, either simultaneously or subsequently. The purpose of this work is to show that the concept of invariant pairs offers a way of representing eigenvalues and eigenvectors that is insensitive to this phenomenon. To demonstrate the use of this concept in the development of numerical methods, we have developed a novel block Newton method for computing such invariant pairs. Algorithmic aspects of this method are considered and a few academic examples demonstrate its viability.     

A computational method for eigenvalues and eigenvectors of a matrix with real eigenvalues

This paper describes a new computational procedure for calculating eigenvalues and eigenvectors of a square matrix. The method is based on a matrix function, the sign of a matrix. Eigenvalues and eigenvectors of matrices with distinct eigenvalues and nondefective matrices with repeated roots can be determined in a straightforward manner. Defective matrices require additional calculations.     

On the eigenvalues of specially low-rank perturbed matrices

We study the eigenvalues of a matrix A perturbed by a few special low-rank matrices. The perturbation is constructed from certain basis vectors of an invariant subspace of A, such as eigenvectors, Jordan vectors, or Schur vectors. We show that most of the eigenvalues of the low-rank perturbed matrix stayed unchanged from the eigenvalues of A; the perturbation can only change the eigenvalues of A that are related to the invariant subspace. Existing results mostly studied using eigenvectors with full column rank for perturbations, we generalize the results to more general settings. Applications of our results to a few interesting problems including the Google’s second eigenvalue problem are presented.     

Computable eigenvalue bounds for rank-k perturbations

We investigate lower bounds for the eigenvalues of perturbations of matrices. In the footsteps of Weyl and Ipsen & Nadler, we develop approximating matrices whose eigenvalues are lower bounds for the eigenvalues of the perturbed matrix. The number of available eigenvalues and eigenvectors of the original matrix determines how close those approximations can be, and, if the perturbation is of low rank, such bounds are relatively inexpensive to obtain. Moreover, because the process need not be restricted to the eigenvalues of perturbed matrices, lower bounds for eigenvalues of bordered diagonal matrices as well as for singular values of rank-k perturbations and other updates of n×m matrices are given.     

Perturbation bounds for means of eigenvalues and invariant subspaces

When a matrix is close to a matrix with a multiple eigenvalue, the arithmetic mean of a group of eigenvalues is a good approximation to this multiple eigenvalue. A theorem of Gershgorin type for means of eigenvalues is proved and applied as a perturbation theorem for a degenerate matrix.For a multiple eigenvalue we derive bounds for computed bases of subspaces of eigenvectors and principal vectors, relating them to the spaces spanned by the last singular vectors of corresponding powers of the matrix. These bounds assure that, provided the dimensionalities are chosen appropriately, the angles of rotation of the subspaces are of the same order of magnitude as the perturbation of the matrix.A numerical example is given.     

The eigenvalues and eigenvectors of finite, low rank perturbations of large random matrices

We consider the eigenvalues and eigenvectors of finite, low rank perturbations of random matrices. Specifically, we prove almost sure convergence of the extreme eigenvalues and appropriate projections of the corresponding eigenvectors of the perturbed matrix for additive and multiplicative perturbation models.The limiting non-random value is shown to depend explicitly on the limiting eigenvalue distribution of the unperturbed random matrix and the assumed perturbation model via integral transforms that correspond to very well-known objects in free probability theory that linearize non-commutative free additive and multiplicative convolution. Furthermore, we uncover a phase transition phenomenon whereby the large matrix limit of the extreme eigenvalues of the perturbed matrix differs from that of the original matrix if and only if the eigenvalues of the perturbing matrix are above a certain critical threshold. Square root decay of the eigenvalue density at the edge is sufficient to ensure that this threshold is finite. This critical threshold is intimately related to the same aforementioned integral transforms and our proof techniques bring this connection and the origin of the phase transition into focus. Consequently, our results extend the class of ‘spiked’ random matrix models about which such predictions (called the BBP phase transition) can be made well beyond the Wigner, Wishart and Jacobi random ensembles found in the literature. We examine the impact of this eigenvalue phase transition on the associated eigenvectors and observe an analogous phase transition in the eigenvectors. Various extensions of our results to the problem of non-extreme eigenvalues are discussed.     

三对角矩阵的特征值及其应用

给出了计算一种三对角矩阵的特征值和特征向量的公式.利用矩阵的特征值理论证明了一些三角恒等式,特别是一些与Fibonacci数和第二类Chebyshev多项式有关的三角恒等式.