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  2. Eigenvalues and eigenvectors of the second derivative

    en.wikipedia.org/wiki/Eigenvalues_and...

    Notation: The index j represents the jth eigenvalue or eigenvector. The index i represents the ith component of an eigenvector. Both i and j go from 1 to n, where the matrix is size n x n. Eigenvectors are normalized. The eigenvalues are ordered in descending order.

  3. Eigenvalues and eigenvectors - Wikipedia

    en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors

    A 2×2 real and symmetric matrix representing a stretching and shearing of the plane. The eigenvectors of the matrix (red lines) are the two special directions such that every point on them will just slide on them. The example here, based on the Mona Lisa, provides a simple illustration. Each point on the painting can be represented as a vector ...

  4. 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 ...

  5. Eigendecomposition of a matrix - Wikipedia

    en.wikipedia.org/wiki/Eigendecomposition_of_a_matrix

    Let A be a square n × n matrix with n linearly independent eigenvectors q i (where i = 1, ..., n).Then A can be factored as = where Q is the square n × n matrix whose i th column is the eigenvector q i of A, and Λ is the diagonal matrix whose diagonal elements are the corresponding eigenvalues, Λ ii = λ i.

  6. Spectrum of a matrix - Wikipedia

    en.wikipedia.org/wiki/Spectrum_of_a_matrix

    The eigendecomposition (or spectral decomposition) of a diagonalizable matrix is a decomposition of a diagonalizable matrix into a specific canonical form whereby the matrix is represented in terms of its eigenvalues and eigenvectors. The spectral radius of a square matrix is the largest absolute value of its eigenvalues.

  7. 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.

  8. Defective matrix - Wikipedia

    en.wikipedia.org/wiki/Defective_matrix

    In linear algebra, a defective matrix is a square matrix that does not have a complete basis of eigenvectors, and is therefore not diagonalizable. In particular, an n × n {\displaystyle n\times n} matrix is defective if and only if it does not have n {\displaystyle n} linearly independent eigenvectors. [ 1 ]

  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 ...