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The Minimum Ratio of Two Eigenvalues [PDF]

SIAM Journal on Applied Mathematics, 1976
The first two eigenvalues, $\lambda _1 $ and $\lambda _2 $, of the problem $y'' + \lambda \phi ( x )y = 0$, $y( { \pm \frac{1}{2}} ) = 0$ are considered. The minimum of their ratio ${\lambda _2 / \lambda _1 }$ is sought for $\phi ( x )$ ranging over the class of piecewise continuous functions satisfying the inequalities $0 < a \leqslant \phi ( x ...
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Yuan's alternative theorem and the maximization of the minimum eigenvalue function [PDF]

Journal of Optimization Theory and Applications, 1994
LetA 1 andA 2 be two symmetric matrices of ordern×n. According to Yuan, there exists a convex combination of these matrices which is positive semidefinite, if and only if the functionx∈R n ↦ max {x T A 1 x,x T A 2 x} is nonnegative.
Alberto Seeger, Juan Enrique Martínez-Legaz   +1 more
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Minimum Eigenvalue Separation

1992
Abstract : For over one hundred years, the eigenvalue problem has been investigated by mathematicians, physicists, and engineers. Scientists explored the characterization, location, perturbation and computation of eigenvalues, to name a few topics. This thesis is devoted to the separation of eigenvalues. We will find the minimum gap between eigenvalues
Tzon-Tzer Lu, Beresford N. Parlett
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Pole assignment with minimum eigenvalue differential sensitivity

Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering, 1997
This paper introduces a set of mathematical formulae for calculating the eigenvalue differential sensitivities of the closed-loop state matrix with respect to the open-loop state matrix, input matrix and state feedback matrix. It provides a computational procedure for a robust pole assignment problem.
Lam, J, Tam, HK
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Computation of the minimum eigenvalue for a nonlinear Sturm-Liouville problem

Lobachevskii Journal of Mathematics, 2014
© 2014, Pleiades Publishing, Ltd. A condition for the existence of a minimum eigenvalue corresponding to a positive eigenfunction of the nonlinear eigenvalue problem for an ordinary differential equation is determined. The problem is approximated by a mesh scheme of the finite element method.
Zheltukhin V., Solov’ev S., Solov’ev P., Chebakova V.   +3 more
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Pole assignment with optimality and minimum eigenvalue sensitivity

Proceedings of the Institution of Electrical Engineers, 1975
A method is presented for designing a constant-gain feedback controller for assigning the closed-loop poles of a linear system to specified locations while minimising an objective functional which is a linear combination of a quadratic performance index and an eigenvalue sensitivity functional.
K. Ramar, V. Gourishankar
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Maximum-Minimum Eigenvalue Detection for Cognitive Radio

2007 IEEE 18th International Symposium on Personal, Indoor and Mobile Radio Communications, 2007
Sensing (signal detection) is a fundamental problem in cognitive radio. In this paper, a new method is proposed based on the eigenvalues of the covariance matrix of the received signal. It is shown that the ratio of the maximum eigenvalue to the minimum eigenvalue can be used to detect the signal existence.
Ying-Chang Liang, Yonghong Zeng
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Design of optimal feedback controllers for minimum eigenvalue sensitivity

Optimal Control Applications and Methods, 1984
AbstractAn iterative method for designing an optimal constant gain feedback controller for a linear system to achieve minimum eigenvalue sensitivity to parameter variations is presented. In addition to assigning eigenvalues to desired locations in the complex plane, one can also assign elements of eigenvectors by this method.
V. G. Gourishankar, Haiming Qiu
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Minimum Properties of Eigenvalues — Elementary Proofs

1980
The purpose of this paper is to use an elementary integral inequality and some simple linear algebra to give a completely elementary proof of the minimum properties of all eigenvalues of Sturm-Liouville problems. The results are a simplification of work published in [1], where singular cases were considered but general boundary conditions were not.
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