Topic
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.
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TL;DR: In this article, the problem of locating the eigenvalues of A relative to S can be transformed into that for an appropriate block companion matrix relative to R by a rational transformation.
Abstract: Let S be a region which can be mapped into a standard region R of the complex plane (unit circle or half-plane) by a rational transformation. Then the problem of locating the eigenvalues of A relative to S can be transformed into that for an appropriate block companion matrix relative to R .
9 citations
TL;DR: In this paper, the bound-state eigenfunctions and eigenvalues of a Schrodinger Hamiltonian are determined as functions of the strength of the potential and the method is able to determine the bound state energies for arbitrarily weak strengths of the possible potential.
Abstract: We present a simply applied numerical technique that allows the accurate determination of the bound-state eigenfunctions and eigenvalues of a differential operator such as the one-particle Schrodinger Hamiltonian. The method applies for potentials that asymptotically vanish. The eigenvalues and eigenfunctions are determined as functions of the strength of the potential and the method is able to determine the bound-state energies for arbitrarily weak strengths of the potential. At no point is a matrix diagonalized thus the method may be applied to problems with space dimension greater than unity.
9 citations
TL;DR: A simple and efficient approach to compute the eigenvalues of the fourth-order Sturm–Liouville equations with variable coefficients by transforming the governing differential equation to a system of algebraic equation based on the polynomial expansion and integral technique.
9 citations
TL;DR: This paper bounds the smallest nonzero eigenvalue, which serves as an indicator of how difficult it is to correctlycompute the desired null space of the approximate alignment matrix.
Abstract: The alignment algorithm of Zhang and Zha is an effective method recently proposed for nonlinear manifold learning (or dimensionality reduction). By first computing local coordinates of a data set, it constructs an alignment matrix from which a global coordinate is obtained from its null space. In practice, the local coordinates can only be constructed approximately and so is the alignment matrix. This together with roundoff errors requires that we compute the the eigenspace associated with a few smallest eigenvalues of an approximate alignment matrix. For this purpose, it is important to know the first nonzero eigenvalue of the alignment matrix or a lower bound in order to computationally separate the null space. This paper bounds the smallest nonzero eigenvalue, which serves as an indicator of how difficult it is to correctly compute the desired null space of the approximate alignment matrix.
9 citations
TL;DR: In this paper, the spectrum structure of discrete second-order Neumann boundary value problems with sign-changing weight was studied, and it was shown that the spectrum consists of real and simple eigenvalues.
Abstract: We study the spectrum structure of discrete second-order Neumann boundary value
problems (NBVPs) with sign-changing weight. We apply the properties of characteristic determinant of the
NBVPs to show that the spectrum consists of real and simple eigenvalues; the number of positive eigenvalues is
equal to the number of positive elements in the weight function, and the number of negative eigenvalues
is equal to the number of negative elements in the weight function. We also show that the eigenfunction
corresponding to the th positive/negative eigenvalue changes its sign exactly times.
9 citations