Showing papers on "Spectrum of a matrix published in 1990"
TL;DR: A continuously differentiable symmetric matrix function of real parameters is constructed that obeys the inequality of the following type: For α ≥ 1, β ≥ 1 using LaSalle's inequality.
Abstract: Optimization problems involving eigenvalues arise in many applications. Let x be a vector of real parameters and let $A(x)$ be a continuously differentiable symmetric matrix function of x. We consi...
218 citations
TL;DR: In this article, the mean of the multiple eigenvalues (in terms of the trace operator tr), the corresponding invariant subspaces, and their derivatives are studied, using a generalization of Sun's approach.
Abstract: The eigenvalues and eigenvectors as well as their derivatives of matrices depending on one or several parameters have been studied by Sun [J. Comput. Math., 3 (1985), pp. 351–364] (using the implicit function theorems) and others, but not for the more general multiple eigenvalues. In this paper, the mean of the multiple eigenvalues (in terms of the trace operator tr), the corresponding invariant subspaces, and their derivatives are studied, using a generalization of Sun’s approach. The numerical aspects concerning the computation of such derivatives by direct and iterative methods (e.g., simultaneous and inverse simultaneous iterations) will be discussed briefly. The main results in this paper apply to clusters of nonmultiple eigenvalues as well. The implications, e.g., on simultaneous iteration, will be explored.
56 citations
TL;DR: In this paper, the distance between the roots of two polynomials in terms of their coefficients and the distances between the eigenvalues of two matrices in the norm of their difference were derived.
47 citations
42 citations
TL;DR: In this paper, necessary and sufficient conditions for a set of numbers to be the eigenvalues of a completion of a matrix prescribed in its upper triangular part are given, where the conditions depend on the number of numbers in the matrix.
Abstract: We give necessary and sufficient conditions for a set of numbers to be the eigenvalues of a completion of a matrix prescribed in its upper triangular part.
31 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 article, upper bounds for the solution of the Lyapunov matrix equation are presented for single eigenvalues, summations of eigen values including the trace, and products of Eigenvalues including the determinant.
Abstract: Eigenvalue upper bounds for the solution of the Lyapunov matrix equation are presented for single eigenvalues, summations of eigenvalues including the trace, and products of eigenvalues including the determinant. These bounds are compared to those for the trace and the extreme eigenvalues. >
22 citations
TL;DR: In this paper, the authors show that the eigenvalues of a graded matrix tend to share the graded structure of the matrix and give precise conditions insuring that this tendency is realized.
Abstract: This paper concerns two closely related topics: the behavior of the eigenvalues of graded matrices and the perturbation of a nondefective multiple eigenvalue. We will show that the eigenvalues of a graded matrix tend to share the graded structure of the matrix and give precise conditions insuring that this tendency is realized. These results are then applied to show that the secants of the canonical angles between the left and right invariant of a multiple eigenvalue tend to characterize its behavior when its matrix is slightly perturbed.
18 citations
TL;DR: In this article, a second-order response-surface model is used to approximate the relationship between a response variable and a set of explanatory variables, and a confidence region around these eigenvalues can be used to aid in the characterization of the stationary point and in an improved ridge analysis of the response surface.
Abstract: A second-order response-surface model is often used to approximate the relationship between a response variable and a set of explanatory variables. The nature of the stationary point of the surface can be determined by considering the eigenvalues of the matrix of the model's second-order terms. Since the elements of this matrix are estimated from the data, however, it follows that the eigenvalues are random variables. Hence the sampling properties of these eigenvalues should be considered in characterizing the nature of the stationary point. In this article, it is shown how a confidence region around these eigenvalues can be used to aid in the characterization of the stationary point and in an improved ridge analysis of the response surface. The delta method is used to construct an approximate confidence region for these eigenvalues. Box and Draper (1987) gave a result that simplifies the calculation of such a confidence region for rotatable or nearly rotatable designs. A simulation study was performed to...
16 citations
TL;DR: In this paper, a simple method is presented to compute the eigenvalues and the Eigenfunctions of second-order linear differential operators, with homogeneous boundary conditions, both in a finite interval and on the semi-line.
Abstract: In this paper a simple method is presented to compute the eigenvalues and the eigenfunctions of second-order linear differential operators, with homogeneous boundary conditions, both in a finite interval and on the semi-line. Our technique overcomes the drawbacks of the method proposed by Calogero to compute the eigenvalues of Sturm-Liouville problems in a finite interval. An estimate of the convergence for the eigenvalues is given in the finite case and numerical tests are performed, exhibiting a very fast rate of convergence for the eigenvalues both for the finite interval and the semi-line cases. An excellent convergence for the eigenfunctions is also obtained in both cases.
14 citations
05 Dec 1990
TL;DR: In this paper, robustness bounds for linear systems with time-invariant uncertainty are studied and the emphasis is on keeping the eigenvalues of the perturbed system in a specified region in the complex plane (such as a wedge).
Abstract: Robustness bounds for linear systems with time-invariant uncertainty are studied. The emphasis is on keeping the eigenvalues of the perturbed system in a specified region in the complex plane (such as a wedge). Using standard Lyapunov techniques, bounds on the uncertain parameters are obtained that guarantee the system eigenvalues to remain in the desired region. >
01 Jan 1990
TL;DR: Lower and upper bounds on the absolute values of the eigenvalues of an n × n real symmetric matrix A are given by (trace A m ) 1/ m for both negative and positive even m .
Abstract: Lower and upper bounds on the absolute values of the eigenvalues of an n × n real symmetric matrix A are given by (trace A m ) 1/ m for both negative and positive even m . (The bounds are within a factor of 2 from the eigenvalues already for m > log 2 n .) We present algorithms for computing trace A m by means of the inversion of some auxiliary matrices of the form λI - A , and we estimate the solution cost for the important special classes of matrices (Toeplitz and Toeplitz-like, banded and sparse having small separator families). The cost is substantially lower than in the approach based on the powering of A . The resulting computation of the eigenvalue bounds is deterministic (it does not depend on the choice of an auxiliary vector as is the case for the power and inverse power methods).
TL;DR: In this article, two theorems on singular values and eigenvalues are given. But they do not consider singular value and Eigenvalue singularity in the same model.
Abstract: (1990). Two Theorems on Singular Values and Eigenvalues. The American Mathematical Monthly: Vol. 97, No. 1, pp. 47-50.
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
07 May 1990
TL;DR: In this article, the theory of eigenvalues for the asymmetric scattering matrix was developed and a method for finding the eigen values of such a matrix was proposed for determining the optimal polarization for radar reflection.
Abstract: The theory of eigenvalues is developed for the asymmetric scattering matrix. The basic properties of eigenvalues are studied. A method is proposed for finding the eigenvalues of such a matrix. It can be proved that the maximum of the magnitude of the eigenvalue is smaller than the maximum of the singular value for the asymmetric scattering matrix. It is shown that the matrix can be diagonalized in two different ways. It is shown that E.M. Kennaugh's (1952) optimal polarization will involve the singular value problem of the asymmetric scattering matrix. On the basis of the spectral theory of matrices, Kennaugh's optimal polarization for radar reflection can be found easily. >
TL;DR: In this paper, the spectrum of abstract kinetic operator A defined by on 0 < x < a with φ(0) = φ (a) is studied, where T and B are self adjoint operators on a Hilbert space.
Abstract: In this paper the spectrum of abstract kinetic operator A defined by on 0 < x < a with φ(0) = φ(a) is studied, where T and B are self adjoint operators on a Hilbert space. The location of non-real eigenvalues and the number of isolated real eigenvalues are estimated. In particular, we investigate the spectrum of the slab transport operator in an inhomogeneous medium.
TL;DR: Perturbation theory for the generalized eigenvalue problem is analyzed in this article, where methods are proposed for regularizing well-posed extremal eigenvalues when the problem itself is almost singular.
Abstract: Perturbation theory for the generalized eigenvalue problem is analysed. Methods are proposed for regularizing well-posed extremal eigenvalues when the problem itself is almost singular. An example is given of the use of regularization in variational problems of mathematical physics.