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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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01 Jan 2011
TL;DR: In this paper, a limit distribution for spacing of random eigenvalues was obtained for the eigenvalue distribution of random values in the Euclidean space. But the limit distribution was not defined.
Abstract: A limit distribution for spacing of random eigenvalues is obtained. The eigenvalues arise �

4 citations

01 Jan 2010
TL;DR: In this paper, the spectral properties of the interior transmission problem have been investigated in inverse scattering theory and lower and upper bounds for the first transmission eigenvalue in terms of the geometry and physical properties of a scattering object were provided.
Abstract: The interior transmission problem arises in inverse scattering theory for inhomogeneous media. It is a boundary value problem for a set of equations defined in a bounded domain coinciding with the support of the scattering object. Of particular interest is the spectrum associated with this boundary value problem, more specifically the existence of eigenvalues known as transmission eigenvalues. Indeed, on one hand, in the context of sampling methods for reconstructing the support of the scatterer one needs to avoid those frequencies that correspond to transmission eigenvalues, and hence it is important to know that the transmission eigenvalues form a discrete set. On the other hand, one can use transmission eigenvalues to obtain information about physical properties of the scattering medium [1], [3] and therefore it is important to know whether they exist and to understand their connection with the index of refraction. The latter application is based on the recent results in [2] which justify the numerical observation that the transmission eigenvalues can be computed from the far field data. Either way, the investigation of the spectral properties of the interior transmission problem has become an interesting mathematical question in inverse scattering theory. We present here the most recent developments on transmission eigenvalues, in particular we show the existence of infinitely many transmission eigenvalues and provide lower and upper bounds for the first transmission eigenvalue in terms of the geometry and physical properties of the scattering object [4], [5]. Then, we show how to use these bounds to obtain information on the index of refraction of a general anisotropic scattering medium, as well on the presence of defects inside the scattering medium.

4 citations

Journal ArticleDOI
TL;DR: The asymptotic eigenvalues of noise covariance matrices in 2-D and 3-D attenuating media are derived and potentially could be used to retrieve medium attenuation properties from observations of noise.
Abstract: Covariance matrices of noise models are used in signal and array processing to study the effect of various noise fields and array configurations on signals and their detectability. Here, the asymptotic eigenvalues of noise covariance matrices in 2-D and 3-D attenuating media are derived. The asymptotic eigenvalues are given by a continuous function, which is the Fourier transform of the infinite sequence formed by sampling the spatial coherence function. The presence of attenuation decreases the value of the large eigenvalues and raises the value of the smaller eigenvalues (compared to the attenuation free case). The eigenvalue density of the sample covariance matrix also shows variation in shape depending on the attenuation, which potentially could be used to retrieve medium attenuation properties from observations of noise.

4 citations

Journal ArticleDOI
TL;DR: In this article, the structure of the one-particle reduced density matrix when expressed in a Cartesian Gaussian basis set is investigated and a set of exact linear dependency conditions between products of basis functions, which result from the angular behaviour of the basis functions is discovered.
Abstract: The structure of the one-particle reduced density matrix when expressed in a Cartesian Gaussian basis set is investigated. A set of exact linear dependency conditions between products of basis functions, which result from the angular behaviour of the basis functions, is discovered. Some of these exact linear dependencies hold simultaneously in both position and momentum spaces making it possible to alter the one matrix while keeping both the position and momentum densities fixed. The magnitude of this space is easily predicted for the Pople and Dunning-Hay basis sets commonly used in quantum chemical calculations, and we give simple rules for their enumeration. It is further shown that alteration of the one-matrix component in this space alters the eigenvalue structure of the one-matrix and therefore has consequences for N-representability. Using the one-matrix corresponding to a wavefunction as a starting point, the eigenvalue change is always in the same direction, that is small eigenvalues get more negative while large ones become more positive. For independent particle model wavefunctions, which are already extreme in their eigenvalues, no change is possible without breaking the N-representability conditions.

4 citations

Journal ArticleDOI
TL;DR: In this article, a method for the calculation of quantum partition functions, and bound eigenvalues and eigenfunctions of the Hamiltonian operator is presented, based on the discretization of the transfer matrix that relates the Feynman path integral to the conventional operator formulation of quantum mechanics.
Abstract: A method for the calculation of quantum partition functions, and bound eigenvalues and eigenfunctions of the Hamiltonian operator is presented. The method is based on the discretization of the transfer matrix that relates the Feynman path integral to the conventional operator formulation of quantum mechanics. Its implementation is very simple, only requiring the diagonalization of the discretized transfer matrix. The method is applied to the harmonic oscillator and Morse potential. The results are in excellent agreement with the exact ones

4 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20238
20229
20202
20193
20187
201731