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  2. Eigenvalues and eigenvectors - Wikipedia

    en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors

    Eigenvalues and eigenvectors are often introduced to students in the context of linear algebra courses focused on matrices. [22] [23] Furthermore, linear transformations over a finite-dimensional vector space can be represented using matrices, [3] [4] which is especially common in numerical and computational applications. [24]

  3. Eigendecomposition of a matrix - Wikipedia

    en.wikipedia.org/wiki/Eigendecomposition_of_a_matrix

    In power iteration, for example, the eigenvector is actually computed before the eigenvalue (which is typically computed by the Rayleigh quotient of the eigenvector). [11] In the QR algorithm for a Hermitian matrix (or any normal matrix), the orthonormal eigenvectors are obtained as a product of the Q matrices from the steps in the algorithm ...

  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. QR algorithm - Wikipedia

    en.wikipedia.org/wiki/QR_algorithm

    The vector converges to an eigenvector of the largest eigenvalue. Instead, the QR algorithm works with a complete basis of vectors, using QR decomposition to renormalize (and orthogonalize). For a symmetric matrix A , upon convergence, AQ = QΛ , where Λ is the diagonal matrix of eigenvalues to which A converged, and where Q is a composite of ...

  6. Rayleigh quotient iteration - Wikipedia

    en.wikipedia.org/wiki/Rayleigh_quotient_iteration

    Rayleigh quotient iteration is an eigenvalue algorithm which extends the idea of the inverse iteration by using the Rayleigh quotient to obtain increasingly accurate eigenvalue estimates. Rayleigh quotient iteration is an iterative method , that is, it delivers a sequence of approximate solutions that converges to a true solution in the limit.

  7. Arnoldi iteration - Wikipedia

    en.wikipedia.org/wiki/Arnoldi_iteration

    In numerical linear algebra, the Arnoldi iteration is an eigenvalue algorithm and an important example of an iterative method.Arnoldi finds an approximation to the eigenvalues and eigenvectors of general (possibly non-Hermitian) matrices by constructing an orthonormal basis of the Krylov subspace, which makes it particularly useful when dealing with large sparse matrices.

  8. Jacobi eigenvalue algorithm - Wikipedia

    en.wikipedia.org/wiki/Jacobi_eigenvalue_algorithm

    In numerical linear algebra, the Jacobi eigenvalue algorithm is an iterative method for the calculation of the eigenvalues and eigenvectors of a real symmetric matrix (a process known as diagonalization).

  9. 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, =.