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  2. Eigenvalues and eigenvectors - Wikipedia

    en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors

    For a matrix, eigenvalues and eigenvectors can be used to decompose the matrix—for example by diagonalizing it. Eigenvalues and eigenvectors give rise to many closely related mathematical concepts, and the prefix eigen-is applied liberally when naming them:

  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. Eigendecomposition of a matrix - Wikipedia

    en.wikipedia.org/wiki/Eigendecomposition_of_a_matrix

    The eigenvalues are real. The eigenvectors of A −1 are the same as the eigenvectors of A. Eigenvectors are only defined up to a multiplicative constant. That is, if Av = λv then cv is also an eigenvector for any scalar c ≠ 0. In particular, −v and e iθ v (for any θ) are also eigenvectors.

  5. Generalized eigenvector - Wikipedia

    en.wikipedia.org/wiki/Generalized_eigenvector

    Using generalized eigenvectors, a set of linearly independent eigenvectors of can be extended, if necessary, to a complete basis for . [8] This basis can be used to determine an "almost diagonal matrix" J {\displaystyle J} in Jordan normal form , similar to A {\displaystyle A} , which is useful in computing certain matrix functions of A ...

  6. Jacobi eigenvalue algorithm - Wikipedia

    en.wikipedia.org/wiki/Jacobi_eigenvalue_algorithm

    When the eigenvalues (and eigenvectors) of a symmetric matrix are known, the following values are easily calculated. Singular values The singular values of a (square) matrix A {\displaystyle A} are the square roots of the (non-negative) eigenvalues of A T A {\displaystyle A^{T}A} .

  7. Power iteration - Wikipedia

    en.wikipedia.org/wiki/Power_iteration

    In mathematics, power iteration (also known as the power method) is an eigenvalue algorithm: given a diagonalizable matrix, the algorithm will produce a number , which is the greatest (in absolute value) eigenvalue of , and a nonzero vector , which is a corresponding eigenvector of , that is, =.

  8. QR algorithm - Wikipedia

    en.wikipedia.org/wiki/QR_algorithm

    In numerical linear algebra, the QR algorithm or QR iteration is an eigenvalue algorithm: that is, a procedure to calculate the eigenvalues and eigenvectors of a matrix.The QR algorithm was developed in the late 1950s by John G. F. Francis and by Vera N. Kublanovskaya, working independently.

  9. Quadratic eigenvalue problem - Wikipedia

    en.wikipedia.org/wiki/Quadratic_eigenvalue_problem

    Quadratic eigenvalue problems arise naturally in the solution of systems of second order linear differential equations without forcing: ″ + ′ + = Where (), and ,,.If all quadratic eigenvalues of () = + + are distinct, then the solution can be written in terms of the quadratic eigenvalues and right quadratic eigenvectors as