The Eigenvalue Problem: Perturbation Theory The Unsymmetric Eigenvalue Problem
نویسنده
چکیده
The Unsymmetric Eigenvalue Problem Just as the problem of solving a system of linear equations Ax = b can be sensitive to perturbations in the data, the problem of computing the eigenvalues of a matrix can also be sensitive to perturbations in the matrix. We will now obtain some results concerning the extent of this sensitivity. Suppose that A is obtained by perturbing a diagonal matrix D by a matrix F whose diagonal entries are zero; that is, A = D + F . If is an eigenvalue of A with corresponding eigenvector x, then we have (D − I)x + Fx = 0. If is not equal to any of the diagonal entries of A, then D − I is nonsingular and we have x = −(D − I)−1Fx. Taking ∞-norms of both sides, we obtain ∥x∥∞ = ∥(D − I)Fx∥∞ ≤ ∥(D − I)F∥∞∥x∥∞,
منابع مشابه
Evolution of the first eigenvalue of buckling problem on Riemannian manifold under Ricci flow
Among the eigenvalue problems of the Laplacian, the biharmonic operator eigenvalue problems are interesting projects because these problems root in physics and geometric analysis. The buckling problem is one of the most important problems in physics, and many studies have been done by the researchers about the solution and the estimate of its eigenvalue. In this paper, first, we obtain the evol...
متن کاملUnsymmetric Wrinkling of Circular Plates
The branching of un,symmetric equilibrium states from axisymmetric equilibrium states for clamped circular plates subjected to a uniform edge thrust and a uniform lateral pressure is analyzed in this paper. The branching process is called wrinkling and the loads at which branching occurs are called wrinkling loads. The nonlinear von Karman plate theory is employed. The wrinkling loads are deter...
متن کاملAnalysis of Natural Frequencies for a Laminated Composite Plate with Piezoelectric Patches using the First and Second Eigenvalue Derivatives
In this paper, the first and second order approximations of Taylor expansion are used for calculating the change of each natural frequency by modifying an arbitrary parameter of a system with a known amount and based on this approximation, the inverse eigenvalue problem is transformed to a solvable algebraic equation. The finite element formulation, based on the classical laminated plate theory...
متن کاملSturm-Liouville Fuzzy Problem with Fuzzy Eigenvalue Parameter
This study is on the fuzzy eigenvalues and fuzzy eigenfunctions of the Sturm-Liouville fuzzy problem with fuzzy eigenvalue parameter. We find fuzzy eigenvalues and fuzzy eigenfunctions of the problem under the approach of Hukuhara differentiability. We solve an example. We draw the graphics of eigenfunctions. We show that eigenfunctions are valid fuzzy functions or not.
متن کاملOn the nonnegative inverse eigenvalue problem of traditional matrices
In this paper, at first for a given set of real or complex numbers $sigma$ with nonnegative summation, we introduce some special conditions that with them there is no nonnegative tridiagonal matrix in which $sigma$ is its spectrum. In continue we present some conditions for existence such nonnegative tridiagonal matrices.
متن کامل