考虑弹性力学中一类上三角无穷维 Hamilton 算子.首先,给出此类Hamilton算子特征值的几何重数和代数指标,进而得到代数重数.其次,根据Hamilton算子特征值的代数重数确定其特征(根)向量组完备的形式,得到此类Hamilton算子特征(根)向量组的完备性是由内部算子特征向量组决定.最后,将所得结果应用到弹性力学问题中. 相似文献
This work deals with the spectral analysis of set-valued operators from a Banach space X into its dual space X*. The main goal of the paper is to study semicontinuity properties of the spectrum operator.Mathematics Subject Classifications (2000) 47H04, 47H12, 58C40.UMR 2071 Universidad de Chile-CNRS. 相似文献
Derivatives of eigenvalues and eigenvectors with respect to parameters in symmetric quadratic eigenvalue problem are studied. The first and second order derivatives of eigenpairs are given. The derivatives are calculated in terms of the eigenvalues and eigenvectors of the quadratic eigenvalue problem, and the use of state space representation is avoided, hence the cost of computation is greatly reduced. The efficiency of the presented method is demonstrated by considering a spring-mass-damper system. 相似文献
This paper concerns a harmonic projection method for computing an approximation to an eigenpair of a large matrix . Given a target point and a subspace that contains an approximation to , the harmonic projection method returns an approximation to . Three convergence results are established as the deviation of from approaches zero. First, the harmonic Ritz value converges to if a certain Rayleigh quotient matrix is uniformly nonsingular. Second, the harmonic Ritz vector converges to if the Rayleigh quotient matrix is uniformly nonsingular and remains well separated from the other harmonic Ritz values. Third, better error bounds for the convergence of are derived when converges. However, we show that the harmonic projection method can fail to find the desired eigenvalue --in other words, the method can miss if it is very close to . To this end, we propose to compute the Rayleigh quotient of with respect to and take it as a new approximate eigenvalue. is shown to converge to once tends to , no matter how is close to . Finally, we show that if the Rayleigh quotient matrix is uniformly nonsingular, then the refined harmonic Ritz vector, or more generally the refined eigenvector approximation introduced by the author, converges. We construct examples to illustrate our theory.
For an nonnegative matrix , an isomorphism is obtained between the lattice of initial subsets (of ) for and the lattice of -invariant faces of the nonnegative orthant . Motivated by this isomorphism, we generalize some of the known combinatorial spectral results on a nonnegative matrix that are given in terms of its classes to results for a cone-preserving map on a polyhedral cone, formulated in terms of its invariant faces. In particular, we obtain the following extension of the famous Rothblum index theorem for a nonnegative matrix: If leaves invariant a polyhedral cone , then for each distinguished eigenvalue of for , there is a chain of distinct -invariant join-irreducible faces of , each containing in its relative interior a generalized eigenvector of corresponding to (referred to as semi-distinguished -invariant faces associated with ), where is the maximal order of distinguished generalized eigenvectors of corresponding to , but there is no such chain with more than members. We introduce the important new concepts of semi-distinguished -invariant faces, and of spectral pairs of faces associated with a cone-preserving map, and obtain several properties of a cone-preserving map that mostly involve these two concepts, when the underlying cone is polyhedral, perfect, or strictly convex and/or smooth, or is the cone of all real polynomials of degree not exceeding that are nonnegative on a closed interval. Plentiful illustrative examples are provided. Some open problems are posed at the end.