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  2. Eigenvalue perturbation - Wikipedia

    en.wikipedia.org/wiki/Eigenvalue_perturbation

    In mathematics, an eigenvalue perturbation problem is that of finding the eigenvectors and eigenvalues of a system = that is perturbed from one with known eigenvectors and eigenvalues =. This is useful for studying how sensitive the original system's eigenvectors and eigenvalues x 0 i , λ 0 i , i = 1 , … n {\displaystyle x_{0i},\lambda _{0i ...

  3. Weyl's inequality - Wikipedia

    en.wikipedia.org/wiki/Weyl's_inequality

    Therefore, Weyl's eigenvalue perturbation inequality for Hermitian matrices extends naturally to perturbation of singular values. [1] This result gives the bound for the perturbation in the singular values of a matrix M {\displaystyle M} due to an additive perturbation Δ {\displaystyle \Delta } :

  4. Category:Perturbation theory - Wikipedia

    en.wikipedia.org/wiki/Category:Perturbation_theory

    This category deals with topics in perturbation theory and variational principles, which commonly occur in the theory of differential equations, with problems in quantum mechanics forming an important subset thereof.

  5. Bauer–Fike theorem - Wikipedia

    en.wikipedia.org/wiki/Bauer–Fike_theorem

    In mathematics, the Bauer–Fike theorem is a standard result in the perturbation theory of the eigenvalue of a complex-valued diagonalizable matrix.In its substance, it states an absolute upper bound for the deviation of one perturbed matrix eigenvalue from a properly chosen eigenvalue of the exact matrix.

  6. Eigengap - Wikipedia

    en.wikipedia.org/wiki/Eigengap

    In linear algebra, the eigengap of a linear operator is the difference between two successive eigenvalues, where eigenvalues are sorted in ascending order. The Davis–Kahan theorem, named after Chandler Davis and William Kahan , uses the eigengap to show how eigenspaces of an operator change under perturbation . [ 1 ]

  7. Eigendecomposition of a matrix - Wikipedia

    en.wikipedia.org/wiki/Eigendecomposition_of_a_matrix

    A generalized eigenvalue problem (second sense) is the problem of finding a (nonzero) vector v that obeys = where A and B are matrices. If v obeys this equation, with some λ , then we call v the generalized eigenvector of A and B (in the second sense), and λ is called the generalized eigenvalue of A and B (in the second sense) which ...

  8. Rayleigh–Ritz method - Wikipedia

    en.wikipedia.org/wiki/Rayleigh–Ritz_method

    An alternative approach, e.g., defining the normal matrix as = of size , takes advantage of the fact that for a given matrix with orthonormal columns the eigenvalue problem of the Rayleigh–Ritz method for the matrix = = can be interpreted as a singular value problem for the matrix . This interpretation allows simple simultaneous calculation ...

  9. Fredholm alternative - Wikipedia

    en.wikipedia.org/wiki/Fredholm_alternative

    For each λ ∈ R, either λ is an eigenvalue of K, or the operator K − λ is bijective from X to itself. Let us explore the two alternatives as they play out for the boundary-value problem. Suppose λ ≠ 0. Then either (A) λ is an eigenvalue of K ⇔ there is a solution h ∈ dom(L) of (L + μ 0) h = λ −1 h ⇔ –μ 0 +λ −1 is an ...

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