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Showing papers on "Spectrum of a matrix published in 1993"


Journal ArticleDOI
TL;DR: In this article, sensitivity analysis of eigenvalues of nonsymmetric operators that depend on parameters is studied, where sensitivity analysis is based on the perturbation method of the eigen values and eigenvectors.
Abstract: This paper is devoted to sensitivity analysis of eigenvalues of nonsym-metric operators that depend on parameters. Special attention is given to the case of multiple eigenvalues. Due to the nondifferentiability (in the common sense) of multiple roots, directional derivatives of eigenvalues and eigenvectors in parametric space are obtained. Sensitivity analysis is based on the perturbation method of eigenvalues and eigenvectors. The generalized eigenvalue problem and vibrational systems are also investigated. Strong and weak interaction of eigenvalues are distinguished and interactions in two- and three-dimensional space are treated geometrically. It is shown that the strong interaction of eigenvalues is a typical catastrophe. Simple examples that illustrate the main ideas are presented. The results obtained are important for qualitative and quantitative study of mechanical systems subjected to static and dynamic instability phenomena.

111 citations


Journal ArticleDOI
TL;DR: This work examines and develops techniques for obtaining a few selected eigenvalues of the generalized eigenvalue problem Ax = [lambda]Bx, where A and B are n[times]n, nonsymmetric, banded complex matrices, and outlines a procedure to separate the converged eigen values from spurious approximations.

53 citations


Journal ArticleDOI
TL;DR: The design and development of a code to calculate the eigenvalues of a large sparse real unsymmetric matrix that are the rightmost, leftmost, or are of the largest modulus are discussed.
Abstract: This paper discusses the design and development of a code to calculate the eigenvalues of a large sparse real unsymmetric matrix that are the rightmost, leftmost, or are of the largest modulus. A subspace iteration algorithm is used to compute a sequence of sets of vectors that converge to an orthonormal basis for the invariant subspace corresponding to the required eigenvalues. This algorithm is combined with Chebychev acceleration if the rightmost or leftmost eigenvalues are sought, or if the eigenvalues of largest modulus are known to be the rightmost or leftmost eigenvalues. An option exists for computing the corresponding eigenvectors. The code does not need the matrix explicitly since it only requires the user to multiply sets of vectors by the matrix. Sophisticated and novel iteration controls, stopping criteria, and restart facilities are provided. The code is shown to be efficient and competitive on a range of test problems.

44 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that the trivial vacuum, where the phases vanish, is a minimum of the free energy of the Wilson line at nonzero temperature, and the corresponding partition function for the phases of these eigenvalues was constructed.

37 citations


Journal ArticleDOI
TL;DR: By utilizing the icosahedral symmetry of the molecule the characteristic polynomial is factorized and the eigenvalues obtained in the Huckel problem for carbon-240.

11 citations


Journal ArticleDOI
TL;DR: In this article, the existence of eigenvalues associated with absolutely continuous eigenfunctions for the generalized Neumann-Poincare operator in the case of chord-are curves was established.
Abstract: This paper presents two approaches aiming at an extension of the classical theory of eigenvalues of the Neumann-Poincare operator. In Sect. 2 the existence of eigenvalues associated with absolutely continuous eigenfunctions for the generalized Neumann-Poincare operator Cг 1 in the case of a chord-are curve г is established which is due to the first named author. In Sect. 3 a more general case of quasicircles due to the second author is presented.

6 citations


Proceedings ArticleDOI
15 Dec 1993
TL;DR: In this paper, the Riccati inequality is reduced to an algebraic Lyapunov inequality with a matrix whose eigenvalues are located at the imaginary axis, and necessary conditions for the solvability of such inequalities are derived.
Abstract: Discusses algebraic tests for the solvability of the indefinite linear matrix inequality (LMI) (A*P+PA+Q/B*P+S* PB+S/R)/spl ges/0 which arises in the general LQ problem and in H/sub /spl infin//-control. The author presents a new geometric algorithm which allows one to directly reduce the LMI to a certain algebraic Riccati inequality (ARI). Under a mild regularity assumption the author describes how to further reduce the Riccati inequality to an indefinite Lyapunov inequality with a matrix whose eigenvalues are located at the imaginary axis. Finally, the author derives new general necessary conditions for the solvability of such Lyapunov inequalities and discusses cases under which these conditions are also sufficient. >

5 citations


Journal ArticleDOI
TL;DR: In this paper, the robustness of eigenvalue distribution in specified complementary regions for perturbed systems is investigated and the proposed sufficient conditions guarantee that the same number of values of the perturbed system lie inside the same region as that of the nominal system.
Abstract: In this paper, we present some results on robustness of eigenvalue distribution in specified complementary regions for perturbed systems. If some eigenvalues of the nominal system are located in a specified region, the proposed sufficient conditions guarantee that the same number of eigenvalues of the perturbed system lie inside the same region. The characteristics of a linear time-invariant system are influenced by the eigenvalue location of the system matrix. Due to uncertainty or parameter variation, all mathematical descriptions of dynamic systems are approximate models at best. The effect of uncertainty will move the eigenvalues of a real system away from the designed ones. Therefore, it is significant to guarantee that the same number of eigenvalues of the perturbed system lie inside the same region as that of the nominal system. By the analysis of eigenvalue distribution, we can explore the locations of dominant eigenvalues, specified eigenvalues or even individual eigenvalues of perturbed systems. Consequently, more properties of perturbed systems such as stability margin, performance robustness and so on can be examined. The proposed theorems can be applied to both continuous- and discrete-time systems. In addition, the analysis of stability robustness can be dealt with as a special case in our study. Two examples are given to show the applicability of the proposed theorems. Finally, some conclusions are presented.

5 citations


Proceedings ArticleDOI
28 Jun 1993
TL;DR: In this article, the eigenvalues are expressed explicitly in terms of the mutual coupling and correlation coefficient in the presence of two sources, and simulation shows that mutual coupling between elements improves the adaptive array's eigenvalue behavior.
Abstract: In adaptive arrays using control loops, the rate of convergence is governed by eigenvalues of the covariance matrix. In cases where the eigenvalues are widely different, the convergence is very slow. In the present work, the eigenvalues are expressed explicitly in terms of the mutual coupling and correlation coefficient in the presence of two sources. Simulation shows that the mutual coupling between elements improves the eigenvalue behavior, and that the improvement is maximum for the adaptive array with about half-wavelength spacing. >

4 citations




01 Jan 1993
TL;DR: In this article, a minimum principle is established for the radial Dirac Hamiltonian for any potential, which uses an r-dependent unitary transformation to decouple the equations for the large and small components of the radial wavefunction; the transformed equation maps to an ordinary Sturm-Liouville equation whose mini- _- mum principle ensures convergence of the eigenvalues from above.
Abstract: A minimum principle is established for the radial Dirac Hamiltonian for any potential. This principle uses an r-dependent unitary transformation to decouple the equations for the large and small components of the radial wavefunction; the transformed equation maps to an ordinary Sturm-Liouville equation whose mini- _- mum principle ensures convergence of the eigenvalues from above. As a concrete and typical example of the application of the principle, basis sets are developed for the Coulomb potential; these sets may be built out of any complete sequence of functions. The positive matrix eigenvalues converge from above to the exact bound-state eigenvalues, the negative eigenvalues converge from below to -mc2, and the wavefunctions corresponding to positive eigenvalues converge in mean- square to the exact bound-state wavefunctions. For the Coulomb potential only, bases of relativistic Sturmian functions are found in which the matrix eigenvalue problem is banded instead of full, and can be solved quickly and stably on a com- - puter even for as many as 4800 basis vectors. An analytic formula is given which expresses the eigenvalues and eigenvectors in terms of the Pollaczek polynomials _ -- an&&eir-zeros. A simple recursion is presented that will evaluate in any Sturmian .

Journal ArticleDOI
TL;DR: In this paper, the controllability grammian of a single input linear time invariant (LTI) system is obtained by computing the controller which assigns the eigenvalues and then solving the Lyapunov equation that defines the grammians.
Abstract: The controllability grammian is important in many control applications. Given a set of closed-loop eigenvalues the corresponding controllability grammian can be obtained by computing the controller which assigns the eigenvalues and then by solving the Lyapunov equation that defines the grammian. The relationship between the controllability grammian, resulting from state feedback, and the closed-loop eigenvalues of a single input linear time invariant (LTI) system is obtained. The proposed methodology does not require the computation of the controller that assigns the specified eigenvalues. The closed-loop system matrix is obtained from the knowledge of the open-loop system matrix, control influence matrix and the specified closed-loop eigenvalues. Knowing the closed-loop system matrix, the grammian is then obtained from the solution of the Lyapunov equation that defines it. Finally the proposed idea is extended to find the state covariance matrix for a specified set of closed-loop eigenvalues (without computing the controller), due to impulsive input in the disturbance channel and to solve the eigenvalue assignment problem for the single input case.

Posted Content
TL;DR: In this paper, the eigenvalues of the Corner Transfer Matrix Hamiltonian associated to the elliptic R matrix of the eight vertex free fermion model are computed in the anisotropic case for magnetic field smaller than the critical value.
Abstract: The eigenvalues of the Corner Transfer Matrix Hamiltonian associated to the elliptic R matrix of the eight vertex free fermion model are computed in the anisotropic case for magnetic field smaller than the critical value. An argument based on generating functions is given, and the results are checked numerically. The spectrum consists of equally spaced levels. IMAFF 93/13

Posted Content
TL;DR: In this article, the authors investigated nonperturbative aspects of zero-dimensional matrix models and showed that the tunneling of eigenvalues correspond to a chaotic sequence of recursion coefficients determining the orthogonal polynomials.
Abstract: I investigate non-perturbative aspects of zero-dimensional matrix models. Subtleties in the large-N limit of the semiclassical picture are pointed out. The tunneling of eigenvalues is seen to correspond to a chaotic sequence of recursion coefficients determining the orthogonal polynomials.