Showing papers on "Spectrum of a matrix published in 1996"
TL;DR: In this article, the eigenvalues and eigenfunctions of a two-level system interacting with a one-mode quantum field are calculated numerically using the operator method.
Abstract: Accurate eigenvalues and eigenfunctions of a two-level system interacting with a one-mode quantum field are calculated numerically. A special iteration procedure based on the operator method permits one to consider the solution within a wide range of the Hamiltonian parameters and to find the uniformly approximating analytical formula for the eigenvalues. Characteristic features of the model are considered, such as the level intersections, the population of the field states and the chaotization in the system through the doubling of the frequencies.
93 citations
07 May 1996
TL;DR: An algorithm based on the properties of the eigenvalues of the covariance matrix of the Wishart matrix that looks for a mismatch between the observed profile and the model in order to detect the presence of a signal.
Abstract: High resolution methods for estimation of parameters in signal processing (bearing angles in array processing or frequencies in spectral analysis for example) can suffer from a bad selection of the model order. This paper proposes an algorithm based on the properties of the eigenvalues of the covariance matrix. In the noise only case, this matrix is a Wishart matrix. For white noise the profile of ordered eigenvalues fits an exponential law. The proposed algorithm uses this property and looks for a mismatch between the observed profile and the model in order to detect the presence of a signal. Under estimation may result from the occurrence of small signal eigenvalues. Performances is greatly improved by the use of deflation for recursive detection-estimation test. Results of simulations are provided in order to show the capabilities of the algorithm.
46 citations
TL;DR: In this paper, it was shown that the rank-one self-adjoint case is irreducible, and a necessary and sufficient condition for the eigenvalues of P and D to interlace is given.
39 citations
TL;DR: In this article, the relative distance between the eigenvalues of a Hermitian matrix and some part of the spectrum of an orthonormal matrix has been investigated, and it is shown that the distance depends on the angle between the ranges of X and HX.
19 citations
TL;DR: In this article, the maximum dimension of a linear space of n × n complex matrices with the property that the cardinality of the spectrum of every A ∈ V is at most k was determined.
8 citations
TL;DR: In this article, it was shown that block random matrices consisting of Wigner-type blocks have as many large (structural) eigenvalues as diagonal blocks, and the asymptotics of the eigen values is sharpened.
6 citations
01 Jan 1996
TL;DR: In this article, the Laplace operator Δ g on a non-compact hyperbolic surface M having a finite area, g the metric is considered and it is well known that the self-adjoint realization of Δ g has a continuous spectrum which is the half-line: (−∞, − 1/4).
Abstract: It is well known that in a proper setup eigenvalues which belong to the discrete spectrum are stable. This property is the basis of the perturbation theory for such eigenvalues. On the other hand, the behavior of eigenvalues which are embedded in the continuous spectrum is completely different. Such eigenvalues may be very unstable under perturbations. A striking example of instability of embedded eigenvalues was given by Colin de Verdiere [4]. Consider the Laplace operator Δ g on a non-compact hyperbolic surface M having a finite area, g the metric. It is well known that the self-adjoint realization of Δ g (denoted by the same symbol) has a continuous spectrum which is the half-line: (−∞, − 1/4]. The discrete spectrum of Δg is a finite set. However, there are many interesting examples where Δ g has infinitely many eigenvalues embedded in the continuous spectrum. That these examples are exceptional in some sense is shown by the following theorem due to Colin de Verdiere.
6 citations
TL;DR: In this article, the authors studied the eigenvalues of Hermitian block Topelitz matrices and showed that the spectrum of any topelitz matrix Pn−1Tn is contained in the interval [r, R], wherer is the smallest eigenvalue andR the largest eigen value of p−1f.
Abstract: We study the asymptotic behaviour of the eigenvalues of Hermitiann×n block Topelitz matricesTn, withk×k blocks, asn tends to infinity. No hypothesis is made concerning the structure of the blocks. Such matrices{Tn} are generated by the Fourier coefficients of a Hermitian matrix valued functionf∈L2, and we study the distribution of their eigenvalues for largen, relating their behaviour to some properties of the functionf. We also study the eigenvalues of the preconditioned matrices{Pn−1Tn}, where the sequence{Pn} is generated by a positive definite matrix valued functionp. We show that the spectrum of anyPn−1Tn is contained in the interval [r, R], wherer is the smallest andR the largest eigenvalue ofp−1f. We also prove that the firstm eigenvalues ofPn−1Tn tend tor and the lastm tend toR, for anym fixed. Finally, exact limit values for both the condition number and the conjugate gradient convergence factor for the preconditioned matricesPn−1Tn are computed.
4 citations
TL;DR: This work describes a divide-and-conquer tridiagonalization approach for matrices with repeated eigenvalues and provides a numerically reliable and effective algorithm for computing the eigenvalue decomposition of a symmetric matrix with two numerically distinct eigen values.
Abstract: We describe a divide-and-conquer tridiagonalization approach for matrices with repeated eigenvalues. Our algorithm hinges on the fact that, under easily constructively verifiable conditions, a symmetric matrix with band width $b$ and $k$ distinct eigenvalues must be block diagonal with diagonal blocks of size at most $b k$. A slight modification of the usual orthogonal band-reduction algorithm allows us to reveal this structure, which then leads to potential parallelism in the form of independent diagonal blocks. Compared to the usual Householder reduction algorithm, the new approach exhibits improved data locality, significantly more scope for parallelism, and the potential to reduce arithmetic complexity by close to 50% for matrices that have only two numerically distinct eigenvalues. The actual improvement depends to a large extent on the number of distinct eigenvalues and a good estimate thereof. However, at worst the algorithms behave like a successive band-reduction approach to tridiagonalization. Moreover, we provide a numerically reliable and effective algorithm for computing the eigenvalue decomposition of a symmetric matrix with two numerically distinct eigenvalues. Such matrices arise, for example, in invariant subspace decomposition approaches to the symmetric eigenvalue problem.
3 citations
3 citations
TL;DR: In this article, the spectrum and the Jordan structure of a matrix pencil Az =z−1B+C+zBT has been considered. And the results have been applied to investigation of the spectrum of two-side infinite periodic Jacobi matrices.
Abstract: The spectrum and the Jordan structure of a matrix pencilAz=z−1B+C+zBT has been considered. The results have been applied to investigation of the spectrum of two-side infinite periodic Jacobi matrices.
TL;DR: In this paper, the eigenvalues and eigenfunctions of the Smoluchowski equation were investigated for the case of potentials with N deep wells, and the small parameter δ =kT/V, which measures the ratio of the thermal energy to a typical well depth, was used in connection with the method of matched asymptotic expansion to obtained asymptic approximations to all the Eigenvalues.
Abstract: The eigenvalues and eigenfunctions of the Smoluchowski equation are investigated for the case of potentials withN deep wells. The small parameter δ=kT/V, which measures the ratio of the thermal energy to a typical well depth, is used in connection with the method of matched asymptotic expansion to obtained asymptotic approximations to all the eigenvalues and eigenfunctions. It is found that the eigensolutions fall into two classes, namely (i) the top-of-the-well and (ii) the bottom-of-the-well eigensolutions. The eigenvalues for both classes of solutions are integer multiples of the squqres of the frequencies at the top or bottom of the various wells. The eigenfunctions are, in general, localized to the top or bottom of the corresponding well. The very small eigenvalues require special consideration because the asymptotic analysis is incapable of distinguishing them from the zero eigenvalue with multiplicityN. Another approximation reveals that, in addition to the true zero eigenvalue, there areN-1 eigenvalues of order exp(−δ). The case of other possible multiple eigenvalues is also examined.