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  2. Eigenvalues and eigenvectors - Wikipedia

    en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors

    Similarly, the geometric multiplicity of the eigenvalue 3 is 1 because its eigenspace is spanned by just one vector []. The total geometric multiplicity γ A is 2, which is the smallest it could be for a matrix with two distinct eigenvalues. Geometric multiplicities are defined in a later section.

  3. Eigenvalue algorithm - Wikipedia

    en.wikipedia.org/wiki/Eigenvalue_algorithm

    Given an n × n square matrix A of real or complex numbers, an eigenvalue λ and its associated generalized eigenvector v are a pair obeying the relation [1] =,where v is a nonzero n × 1 column vector, I is the n × n identity matrix, k is a positive integer, and both λ and v are allowed to be complex even when A is real.l When k = 1, the vector is called simply an eigenvector, and the pair ...

  4. Spectrum of a matrix - Wikipedia

    en.wikipedia.org/wiki/Spectrum_of_a_matrix

    Thus the elements of the spectrum are precisely the eigenvalues of T, and the multiplicity of an eigenvalue λ in the spectrum equals the dimension of the generalized eigenspace of T for λ (also called the algebraic multiplicity of λ). Now, fix a basis B of V over K and suppose M ∈ Mat K (V) is a matrix.

  5. Inverse iteration - Wikipedia

    en.wikipedia.org/wiki/Inverse_iteration

    Naively, if at each iteration one solves a linear system, the complexity will be k O(n 3), where k is number of iterations; similarly, calculating the inverse matrix and applying it at each iteration is of complexity k O(n 3). Note, however, that if the eigenvalue estimate remains constant, then we may reduce the complexity to O(n 3) + k O(n 2 ...

  6. Eigenfunction - Wikipedia

    en.wikipedia.org/wiki/Eigenfunction

    This solution of the vibrating drum problem is, at any point in time, an eigenfunction of the Laplace operator on a disk.. In mathematics, an eigenfunction of a linear operator D defined on some function space is any non-zero function in that space that, when acted upon by D, is only multiplied by some scaling factor called an eigenvalue.

  7. Rayleigh–Ritz method - Wikipedia

    en.wikipedia.org/wiki/Rayleigh–Ritz_method

    Truncated singular value decomposition (SVD) in numerical linear algebra can also use the Rayleigh–Ritz method to find approximations to left and right singular vectors of the matrix of size in given subspaces by turning the singular value problem into an eigenvalue problem.

  8. Characteristic polynomial - Wikipedia

    en.wikipedia.org/wiki/Characteristic_polynomial

    In linear algebra, eigenvalues and eigenvectors play a fundamental role, since, given a linear transformation, an eigenvector is a vector whose direction is not changed by the transformation, and the corresponding eigenvalue is the measure of the resulting change of magnitude of the vector.

  9. 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 ...