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: A fast recursive algorithm is developed to construct numerically a matrix with prescribed eigenvalues and singular values of an arbitrary matrix based on the Weyl--Horn theorem.
Abstract: The Weyl--Horn theorem characterizes a relationship between the eigenvalues and the singular values of an arbitrary matrix. Based on that characterization, a fast recursive algorithm is developed to construct numerically a matrix with prescribed eigenvalues and singular values. Besides being of theoretical interest, the technique could be employed to create test matrices with desired spectral features. Numerical experiment shows this algorithm to be quite efficient and robust.
27 citations
TL;DR: In this article, the eigenvalues of the spectral second order derivative matrices were studied and an exponential convergence was shown for the first part of the spectrum and an asymptotic behavior for the other eigen values.
Abstract: We study the eigenvalues of the spectral second order derivative matrices. We prove an exponential convergence for the first part of the spectrum and we give an asymptotic behavior for the other eigenvalues.
27 citations
TL;DR: In this paper, the eigenvalues of tridiagonal lambda-matrices are verified using Sturm sequences, and the results are shown to be real in most cases, as the theorems of this paper show.
26 citations
TL;DR: In this article, the eigenvalues of the poly-Laplacian with any order on a domain in an n-dimensional unit sphere were studied and the optimal result of Cheng and Yang (Math Ann 331:445-460, 2005) was included in their results.
Abstract: In this paper we study eigenvalues of the poly-Laplacian with any order on a domain in an n-dimensional unit sphere and obtain estimates for eigenvalues. In particular, the optimal result of Cheng and Yang (Math Ann 331:445–460, 2005) is included in our ones. In order to prove our results, we introduce 2(l + 1) functions ai and bi, for i = 0, 1, . . . , l and two operators μ and η. First of all, we study properties of functions ai and bi and the operators μ and η. By making use of these properties and introducing k free constants, we obtain estimates for eigenvalues.
26 citations
TL;DR: In this paper, the authors define the set Si,n to be the set of all integers from 0 to n, excluding i, and characterize the structures of the graphs whose Laplacian matrix has such a set as its eigenvalues.
Abstract: In this paper, we investigate graphs for which the corresponding Laplacian matrix has distinct integer eigenvalues. We define the set Si,n to be the set of all integers from 0 to n, excluding i. If there exists a graph whose Laplacian matrix has this set as its eigenvalues, we say that this set is Laplacian realizable. We investigate the sets Si,n that are Laplacian realizable, and the structures of the graphs whose Laplacian matrix has such a set as its eigenvalues. We characterize those i < n such that Si,n is Laplacian realizable, and show that for certain values of i, the set Si,n is realized by a unique graph. Finally, we conjecture that Sn,n is not Laplacian realizable for n ≥ 2 and show that the conjecture holds for certain values of n. © 2005 Wiley Periodicals, Inc. J Graph Theory
26 citations