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Showing papers on "Spectrum of a matrix published in 1976"


Journal ArticleDOI
TL;DR: In this article, a Gaussian probability density function with the same mean and variance is used to calculate the eigenvalue spectrum of a large symmetric square matrix, each of whose upper triangular elements is described by the Gaussian distribution.
Abstract: A new and straightforward method is presented for calculating the eigenvalue spectrum of a large symmetric square matrix each of whose upper triangular elements is described by a Gaussian probability density function with the same mean and variance. Using the n to 0 method, the authors derive the semicircular eigenvalue spectrum when the mean of each element is zero and show that there is a critical finite mean value above which a single eigenvalue splits off from the semicircular continuum of eigenvalues.

187 citations


Journal ArticleDOI
TL;DR: In this paper, a matrix norm condition is given under which the large eigenvalues of a two-time scale system will be sufficiently separated from the small eigen values, and a feedback control design is proposed in which two gain matrices are used for separate placement of small and large values.

66 citations


Book ChapterDOI
01 Jan 1976
TL;DR: The Lanczos algorithm as mentioned in this paper can be used to approximate both the largest and smallest eigenvalues of a symmetric matrix whose order is so large that similarity transformations are not feasible.
Abstract: The Lanczos algorithm can be used to approximate both the largest and smallest eigenvalues of a symmetric matrix whose order is so large that similarity transformations are not feasible. The algorithm builds up a tridiagonal matrix row by row and the key question is when to stop. An analysis leads to a stopping criterion which is inspired by a useful error bound on the computed eigenvalues.

32 citations


Journal ArticleDOI
TL;DR: In this paper, general symmetric hypoelliptic systems of differential operators in with discrete spectrum are considered and two-sided estimates for, the number of eigenvalues in the interval.
Abstract: General symmetric hypoelliptic systems of differential operators in with discrete spectrum are considered. Two-sided estimates, as , are found for , the number of eigenvalues in the interval . Under a regularity assumption on the behavior of the spectrum of the Weyl matrix symbol of the system, these estimates reduce to the asymptotics of with an estimate of the remainder term. In part the results are also new for the scalar case. Bibliography: 9 titles.

24 citations


Journal ArticleDOI
TL;DR: In this article, the problem of determining a general family of output feedback matrices K which drive the eigenvalues of a controllable and observable linear time-invariant system to desired prespecified positions is investigated.
Abstract: This paper investigates the problem of determining a general family of output feedback matrices K which drive the eigenvalues of a controllable and observable linear time-invariant system to desired prespecified positions. The approach followed is based on the concept of making the closed-loop system matrix A + BKC similar to another matrix H of the same dimensions which possesses the desired closed-loop eigenvalues. This approach reduces the problem of determining a general family of K's to that of solving a linear set of equations. In addition, necessary conditions are established for stabilizing and arbitrarily assigning all eigenvalues by output feedback.

14 citations


Journal ArticleDOI
TL;DR: The spectrum of the linear transport operator under generalized boundary conditions proposed by Belleni-Morante (1970 a) contains in the right half-plane (λ|Reλ>−1) only a finite number of real eigenvalues.
Abstract: The spectrum of the linear transport operator under generalized boundary conditions proposed by Belleni-Morante (1970 a) contains in the right half-plane (λ|Reλ>−1) only a finite number of real eigenvalues. Its essential spectrum consists of a ‘fan’ of continuous spectrum lines with apex at λ=−1.

13 citations


Journal ArticleDOI
TL;DR: Asymptotic formulas for the distribution function of the eigenvalues accumulating at the end of a lacuna of the continuous spectrum of the perturbed Hill operator were obtained in this article.
Abstract: Asymptotic formulas are obtained for the distribution function of the eigenvalues accumulating at the end of a lacuna of the continuous spectrum of the perturbed Hill operator.

11 citations


Journal ArticleDOI
TL;DR: In this paper, a numerical technique is presented for locating the eigenvalues of two point linear differential eigenvalue problems, namely, the Orr-Sommerfeld equation of the plane Poiseuille flow.

8 citations



Journal ArticleDOI
TL;DR: In this paper, it was shown that complex eigenvalues actually exist in the half-plane of a single-speed spherical shell surrounding a purely absorbing core and that the eigenvalue spectrum of the neutron transport equation in a finite body is discrete.
Abstract: If the assumption is made that the neutron speed has a positive lower bound, the eigenvalue spectrum of the neutron transport equation in a finite body is discrete1 Let λ∗ be the minimum collision rate Then it can also be shown, under rather broad assumptions, the most relevant of which is the isotropy of scattering, that the eigenvalues belonging to the half-plane Reλ > −λ∗ must be real Real eigenvalues also exist, as a rule, in the remaining half-plane Reλ ≤ −λ∗2 An old problem is: does this half-plane contain complex (ie, nonreal) eigenvalues? In this paper we study a particular one-speed problem (a thin absorbing and scattering spherical shell surrounding a purely absorbing core) and show that such complex eigenvalues actually exist

7 citations


Journal ArticleDOI
TL;DR: In this article, a single-input modal control theory is developed whereby the loop gains of a time-invariant multi-variable system may be calculated using simple formulae for the cases when both the open-loop plant matrix and the closed-loop plants matrix have a number of distinct and confluent eigenvalues.
Abstract: A single-input modal control theory is developed whereby the loop gains of a single-input time-invariant multi-variable system may be calculated using simple formulae for the cases when both the open-loop plant matrix and the closed-loop plant matrix have a number of sets of distinct and confluent eigenvalues. The results may be applied in a sequential manner for multi-input systems with repeated sets of confluent eigenvalues provided that the appropriate mode-controllability conditions are satisfied. A number of illustrative examples are included.

Journal ArticleDOI
TL;DR: For a class of linear distributed-parameter systems formulas and procedures are given by which one can compute the 1st and 2nd-order sensitivity functions of the state feedback gain, which moves the system eigenvalues to desired positions, to variations of the system parameters as mentioned in this paper.
Abstract: For a class of linear distributed-parameter systems formulas and procedures are given by which one can compute the 1st-and 2nd-order sensitivity functions of the state feedback gain, which moves the system eigenvalues to desired positions, to variations of the system parameters. Analogous results are derived for the sensitivities of the closed-loop eigenvalues.

Journal ArticleDOI
TL;DR: The method of calculating the shifts in the eigenvalues of a perturbed matrix is given and the perturbation matrices of same types of perturbations are derived and these equations form the basis of an efficient computer programme that is used in various network esign problems.
Abstract: If a perturbed network or system is described by the sum of the matrix describing the original network and a perturbation matrix, then the shifts in the natural frequencies can be conveniently calculated and studied. In this paper the method of calculating the shifts in the eigenvalues of a perturbed matrix is given and the perturbation matrices of same types of perturbations are derived. The method is then used to formulate a set of simultaneous equations relating the eigenvalue shifts to the variations in the network elements. These equations form the basis of an efficient computer programme that is used in various network esign problems.