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.

W. K. B. Approximation and Scaling

Abstract: It is shown that the W. K. B. eigenvalues and eigenfunctions satisfy the Hellmann-Feynman theorem and the virial theorem. As a consequence the scaling behaviour of the exact eigenvalues with respect to the para­meters in the Hamiltonian is retained in the W. K. B. approximation for the polynomial potentials. Hypervirial relations of certain lower orders are also satisfied by the W. K. B. approximants.

Eigenvalue ratio detection based on exact moments of smallest and largest eigenvalues

Abstract: Detection based on eigenvalues of received signal covariance matrix is currently one of the most effective solution for spectrum sensing problem in cognitive radios. However, the results of these schemes always depend on asymptotic assumptions since the close-formed expression of exact eigenvalues ratio distribution is exceptionally complex to compute in practice. In this paper, non-asymptotic spectrum sensing approach to approximate the extreme eigenvalues is introduced. In this context, the Gaussian approximation approach based on exact analytical moments of extreme eigenvalues is presented. In this approach, the extreme eigenvalues are considered as dependent Gaussian random variables such that the joint probability density function (PDF) is approximated by bivariate Gaussian distribution function for any number of cooperating secondary users and received samples. In this context, the definition of Copula is cited to analyze the extent of the dependency between the extreme eigenvalues. Later, the decision threshold based on the ratio of dependent Gaussian extreme eigenvalues is derived. The performance analysis of our newly proposed approach is compared with the already published asymptotic Tracy-Widom approximation approach.

Eigenvalues of matrix commutators

Abstract: We characterize the eigenvalues of [X,A]=XA−AX, where A is an n by n fixed matrix and X runs over the set of the matrices of the same size.

Eigenvalues of element stiffness matrices. part i: 2‐d plane elements

Abstract: The eigenvalues of element stiffness matrices K and the eigenvalues of the generalized problem Kx = λMx, where M is the element's mass matrix, are of fundamental importance in finite element analysis. For instance, they may indicate the presence of ‘zero energy modes’, or control the critical timestep applicable in temporal integration of dynamic problems. Recently explicit formulae for the eigenvalues of the stiffness matrix of a plane, 4‐node rectangular element have been given, and the authors have extended this approach to deal with 8‐node solid brick elements as well. In the present paper, explicit eigenvalues are given for plane triangular elements and techniques for eigenmode visualization are applied to well‐known triangular and quadrilateral elements. In the companion paper (Part II), the stiffness matrices of solid tetrahedra and bricks are similarly treated.

Auto-Corner Detection Based on the Eigenvalues Product of Covariance Matrices over Multi-Regions of Support

Abstract: In this paper we present an auto-detection corner based on eigenvalues product of covariance matrices (ADEPCM) of boundary points over multi-region of support. The algorithm starts with extracting the contour of an object, and then computes the eigenvalues product of covariance matrices of this contour at various regions of support. Finally determine automatically peaks of the graph of eigenvalues product function. We consider that points corresponding to peaks of eigenvalues product graph are reported as corners, which avoids human judgment and curvature threshold settings. Experimental results show that the proposed method has more robustness for noise and various geometrical transform.