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对于判断矩阵重特征值的存在性问题,运用“若λ是矩阵A的特征值,则入“是Ak的特征值”这一性质,通过矩阵的迹与特征值的关系,得到了实数域上矩阵重特征值的存在性定理并给出了证明.定理实现了“由矩阵幂运算来判断矩阵重特征值的存在性”这样一个计算过程,对讨论矩阵特征值问题具有一定的启示意义. 相似文献
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本对具有k重零特征根的矩阵的一些性质进行了探讨,这些性质主要涉及到矩阵的秩、矩阵的有理标准形和约当标准形等方面。 相似文献
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本文利用2×2阶实对称矩阵特征值的计算,并以秩—1修正为基础,通过建立一种二分模式,得到了计算n除实对称三对角矩阵所有特征值的新方法.结果表明,当要求所有特征值时,本文方法优于QR方法。由于算法过程中数据的不相关性,本文方法具有很好的并行性,尤其适合于MIMD并行实现。 相似文献
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借助相似变换将非亏损矩阵转为Hessenberg矩阵,通过获得确定Hessenberg矩阵特征多项式系数的方法,利用特征值与特征多项式系数间的关系,给出求非亏损矩阵特征值的一种数值算法。 相似文献
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本文将实对称矩阵特征值的交错定理推广到实对称区间矩阵,给出了实对称区间矩阵特征值确界的交错定理,并应用该定理构造了估计实对称三对角区间矩阵特征值界的算法.文中数值例子表明,本文所给算法与一些现有算法相比在使用范围、计算精度和计算量等方面都具有一定的优越性. 相似文献
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对于n阶实对称矩阵A,在不知道某个特征值(不管重数)所对应的特征向量时.我们得出了A的表示式:其中λri是A的ri重特征值p1(λri),…,pri(λri)是λri的特征子空间的正交基底. 相似文献
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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. 相似文献
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矩阵约当标准化的一个新方法 总被引:1,自引:0,他引:1
在线性和非线性问题的研究中,常需要构造一个基,使线性算子T在此基础下的矩阵表示为约当标准型,本文介绍了构造这种基的一个方法,我们从T的每个特征向量开始,通过求解一系列线性方程组而求得广义特征向量的一个链,将所有这种链放在一起,便构成想要的一组基,与通常的方法相比,这一方法较易操作,计算量小。 相似文献
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对称双边对角矩阵特征值问题的计算 总被引:1,自引:0,他引:1
1 引 言 大型稀疏矩阵在工程上有广泛的应用.例如,结构工程的有限元分析、电力系统的分析、流体力学及图像数据压缩等应用中常遇到求大型稀疏矩阵的特征值问题.因而矩阵特征值计算问题成为数值代数领域长期关注的问题,如[6][7].最近M.Gu与S.C.Eisenstat 相似文献
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矩阵特征值、特征向量的确定 总被引:4,自引:1,他引:3
首先对由 A的特征值、特征向量求 A- 1 ,AT,A* ( A的伴随矩阵 )、P- 1 AP以及 A的多项式φ( A)的特征值和特征向量的结论作了个归纳 ;对相反的情形 ,我们给出了部分已有的结果 ,并通过四道例题着重讨论了如何由 φ( A)的特征值来求 A的特征值 . 相似文献
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Alexei A. Mailybaev 《Numerical Linear Algebra with Applications》2006,13(5):419-436
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. 相似文献
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证明了由特征值及特征向量反求矩阵时,特征值在对角矩阵中的排序可以是任意的,只须将对应特征向量作相应排序,所得矩阵唯一。对于重特征值的线性无关的特征向量可任意选取,所得矩阵唯一。 相似文献
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Jordan标准形过渡矩阵求法的补充条件 总被引:2,自引:2,他引:0
用反例证明了用方阵的特征向量逆推Jordan链构作Jordan标准形过渡矩阵的方法在理论上不成立,并给出了使这个方法成立的补充条件. 相似文献
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Christian Mehl Volker Mehrmann André C.M. Ran Leiba Rodman 《Linear algebra and its applications》2012,436(10):4027-4042
For selfadjoint matrices in an indefinite inner product, possible canonical forms are identified that arise when the matrix is subjected to a selfadjoint generic rank one perturbation. Genericity is understood in the sense of algebraic geometry. Special attention is paid to the perturbation behavior of the sign characteristic. Typically, under such a perturbation, for every given eigenvalue, the largest Jordan block of the eigenvalue is destroyed and (in case the eigenvalue is real) all other Jordan blocks keep their sign characteristic. The new eigenvalues, i.e. those eigenvalues of the perturbed matrix that are not eigenvalues of the original matrix, are typically simple, and in some cases information is provided about their sign characteristic (if the new eigenvalue is real). The main results are proved by using the well known canonical forms of selfadjoint matrices in an indefinite inner product, a version of the Brunovsky canonical form and on general results concerning rank one perturbations obtained. 相似文献
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We present methods for computing a nearby partial Jordan-Schur form of a given matrix and a nearby partial Weierstrass-Schur
form of a matrix pencil. The focus is on the use and the interplay of the algorithmic building blocks – the implicitly restarted
Arnoldi method with prescribed restarts for computing an invariant subspace associated with the dominant eigenvalue, the clustering
method for grouping computed eigenvalues into numerically multiple eigenvalues and the staircase algorithm for computing the
structure revealing form of the projected problem. For matrix pencils, we present generalizations of these methods. We introduce
a new and more accurate clustering heuristic for both matrices and matrix pencils. Particular emphasis is placed on reliability
of the partial Jordan-Schur and Weierstrass-Schur methods with respect to the choice of deflation parameters connecting the
steps of the algorithm such that the errors are controlled. Finally, successful results from computational experiments conducted
on problems with known canonical structure and varying ill-conditioning are presented.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
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本文给出并论证了 ,当 n阶实方阵 A具有 i ( 1≤ i≤ n)个 (即任意多个 )模最大的特征值时 ,用幂法求出这些模最大的特征值及其相应特征向量的方法 .该方法是对幂法理论的进一步完善 相似文献