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 paper, it was shown that any real matrix with distinct eigenvalues is similar to a real, irreducible, tridiagonal matrix with a matching of size at least k.
1 citations
TL;DR: A procedure for detecting the extreme eigenvalues that relies on the relationship between the acceleration rate of polynomial acceleration iteration and the norm of the matrix via the spectral theorem, applicable to normal matrices is described.
Abstract: For a self-adjoint linear operator with a discrete spectrum or a Hermitian matrix, the ''extreme'' eigenvalues define the boundaries of clusters in the spectrum of real eigenvalues. The outer extreme ones are the largest and the smallest eigenvalues. If there are extended intervals in the spectrum in which no eigenvalues are present, the eigenvalues bounding these gaps are the inner extreme eigenvalues. We will describe a procedure for detecting the extreme eigenvalues that relies on the relationship between the acceleration rate of polynomial acceleration iteration and the norm of the matrix via the spectral theorem, applicable to normal matrices. The strategy makes use of the fast growth rate of Chebyshev polynomials to distinguish ranges in the spectrum of the matrix which are devoid of eigenvalues. The method is numerically stable with regard to the dimension of the matrix problem and is thus capable of handling matrices of large dimension. The overall computational cost is quadratic in the size of a dense matrix; linear in the size of a sparse matrix. We verify computationally that the algorithm is accurate and efficient, even on large matrices.
1 citations
TL;DR: In this article, a Gershgorin's type result on the localisation of the spectrum of a matrix is presented, which relies upon the method of Schur complements, and yields estimates that hold without major differences in the cases of both scalar and operator matrices.
Abstract: We present a Gershgorin's type result on the localisation of the spectrum of a matrix. Our method is elementary as it relies upon the method of Schur complements, but it outperforms the one based on the Cassini ovals of Ostrovski and Brauer. Furthermore, it yields estimates that hold without major differences in the cases of both scalar and operator matrices. Several refinements of known results are obtained.
1 citations
01 Jan 2013
TL;DR: In this article, a self-adjoint linear operator A is defined in a suitable Hilbert space H such that the eigenvalues of such a problem coincide with those of A.
Abstract: In this work, we study a fourth-order boundary value problem problem with eigenparameter dependent boundary conditions and transmission conditions at a interior point. A self-adjoint linear operator A is defined in a suitable Hilbert space H such that the eigenvalues of such a problem coincide with those of A. We obtain asymptotic formulae for its eigenvalues and fundamental solutions. 2010 Mathematics Subject Classification. 34L20, 35R10.
1 citations
01 Jan 1983
TL;DR: In this article, the sensitivity of the eigenvalues of a defective matrix under small perturbations was studied and a generalization of the special results of Wilkinson, Stewart,Bauer and Fike was presented.
Abstract: This paper is concerned with the sensitivity of the eigenvalues of a defective matrix under small perturbations.The given estimate generalizes all special results of Wilkinson, Stewart,Bauer and Fike,when the eigenvalue is simple and when the matrix is nondefective, and interpretes the phenomenon indicated by Golub and Wilkinson for Multiple eigenvalues
1 citations