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, an effective algorithm is provided for determining the number of negative eigenvalues of a one-dimensional Schrodinger operator with point interactions in terms of the intensities and the distances between the interactions.
Abstract: An effective algorithm is provided for determining the number of negative eigenvalues of a one-dimensional Schrodinger operator with point interactions in terms of the intensities and the distances between the interactions
43 citations
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42 citations
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TL;DR: In this paper, it was shown that a matrix with linearly independent eigenvectors is similar to a hermitian matrix, and can consequently be transformed into its conjugate transpose by a positive definite Hermitian similarity.
Abstract: In this note we are, essentially, concerned with generalizations of the (known) fact tha t an n • matr ix with n linearly independent eigenvectors all corresponding to real eigenvalues is similar to a hermitian matrix, and can consequently be transformed into its conjugate transpose by a positive definite hermitian similarity. We first establish, fo r ' any positive integer m, an analogous necessary and sufficient condition tha t a given square complex matr ix A should have a set of real eigenvalues, not necessarily all distinct, to which therecorrespond at least m linearly,independent eigenvectors; this of course implies a corresponding result about pure imaginary eigenvalues. We also obtain an analogous result concerning eigenvalues of modulus unity: As a simple application of our more general results, we establish, in Theorem 4, the reality of the eigenvalues of a certain rather special type of matrix. Throughout, we shall use A = (aij) to denote an arbi t rary n • n complex matrix, and A* will denote the transposed conjugate matr ix; we denote the rank of A by r(A).
42 citations
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42 citations
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TL;DR: In this article, a simple explicit convergence criterion is given, as well as the algorithm and two numerical examples for those eigenvalues of a λ-matrix whose elements are functions of a parameter λ.
Abstract: The matrix N(λ) whose elements are functions of a parameter λ is called the λ-matrix. Those values of λ that make the matrix singular are of great interest in many applied fields. An efficient method for those eigenvalues of a λ-matrix is presented. A simple explicit convergence criterion is given, as well as the algorithm and two numerical examples.
41 citations