Spectrum of a matrix

About: Spectrum of a matrix is a research topic. Over the lifetime, 1064 publications have been published within this topic receiving 19841 citations. The topic is also known as: matrix spectrum.

Implementing Approximations to Extreme Eigenvalues and Eigenvalues of Irregular Surface Partitionings for Use in SAR and CAR Models

Abstract: Good approximations of eigenvalues exist for the regular square and hexagonal tessellations. To complement this situation, spatial scientists need good approximations of eigenvalues for irregular tessellations. Starting from known or approximated extreme eigenvalues, the remaining eigenvalues may be in turn approximated. One reason spatial scientists are interested in eigenvalues is because they are needed to calculate the Jacobian term in the autonormal probability model. If eigenvalues are not needed for model fitting, good approximations are needed to give the interval within which the spatial parameter will lie.

A partitioning method for finding eigenvalues and eigenvectors

Abstract: The need to compute eigenvalues and eigenvectors of very large matrices arises in many areas of science and engineering. One such example is in power systems. The successful operation of a power system requires that the system be stable for a wide range of the operating conditions. The purpose of this research was to study small-signal stability of large power systems. In the presented approach the total system is first divided into subsystems, and the eigenvalues and the eigenvectors of the subsystems are computed using the standard library routines. Each subsystem represents some physical part of the entire system. The subsystems are next interconnected, and the interactions between them are studied. Only the critical eigenvalues are computed. The eigenvalues of the total system are classified as local or global. Local eigenvalues are mainly influenced by one eigenvalue of a subsystem, and the change between the subsystem and total system eigenvalues is small. For global eigenvalues, several eigenvalues of subsystems interact with each other, and the change between subsystem and total system eigenvalues may be large. Eigenvalues of subsystems are examined to determine which ones interact with each other and may move substantially. Those which are found to interact with each other are studied together using a method based on the invariant subspaces. The invariant subspaces method solves for a group of eigenvalues and corresponding eigenvectors at the same time. It is similar to Newton's method, but it overcomes many problems which Newton's method has, such as: convergence to the same final eigenvalue starting from different eigenvalues of subsystems, poor convergence properties in the case of close eigenvalues, and problems with the solution of a large linear equation using iterative methods. The invariant subspaces method is capable of handling cases of close or equal eigenvalues, it has better convergence properties, especially when the global eigenvalues are studied, and the large set of linear equations, generated by it, is easier to solve.

Asymptotic expansions of the largest eigenvalues

Abstract: In this paper, we provide a rigorous derivation of asymptotic formula for the largest eigenvalues using the convergence estimation of the eigenvalues of a sequence of self-adjoint compact operators of perturbations resulting from the presence of small inhomogeneities.

Identifying Coefficients in the Spectral Representation for First Passage Times.

Abstract: : The spectral approach to first passage time distributions for Markov processes requires knowledge of the eigenvalues and eigenvectors of the infinitesimal generator matrix. We demonstrate that in many cases knowledge of the eigenvalues alone is sufficient to compute the first passage time distribution.

Eigenstates of Large Profiled Matrices

Abstract: Publisher Summary The coefficient matrices of large systems of equations frequently have a profile structure. For dynamic and stability analyses, a considerable number of eigenstates of such matrices must be determined, for instance the eigenstates with the smallest eigenvalues. The algorithm of the eigenstate solver should be capable of handling multiple eigenvalues, which are common in symmetric structural systems. Conventional algorithms, such as algorithms based on the method of Lanczos, do not yield the eigenvalues in sequence and require special handling of multiple eigenvalues. This chapter presents a new algorithm for the special eigenvalue problem that preserves convex matrix profiles during iteration, yields the eigenvalues in ascending order, and does not require special precautions for multiple eigenvalues. The algorithm is applied to dam vibration problems. Multiple eigenvalues do not require special treatment. Eigenvalues of equal magnitude, but opposite sign, require special consideration.