enow.com Web Search

Search results

  1. Results from the WOW.Com Content Network
  2. Eigenvalue algorithm - Wikipedia

    en.wikipedia.org/wiki/Eigenvalue_algorithm

    The eigenvalues of a Hermitian matrix are real, since (λ − λ)v = (A * − A)v = (A − A)v = 0 for a non-zero eigenvector v. If A is real, there is an orthonormal basis for R n consisting of eigenvectors of A if and only if A is symmetric. It is possible for a real or complex matrix to have all real eigenvalues without being Hermitian.

  3. Eigenvalues and eigenvectors - Wikipedia

    en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors

    The corresponding eigenvalue, characteristic value, ... The classical method is to first find the eigenvalues, and then calculate the eigenvectors for each eigenvalue.

  4. 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.

  5. Jacobi eigenvalue algorithm - Wikipedia

    en.wikipedia.org/wiki/Jacobi_eigenvalue_algorithm

    Thus one can only calculate the numerical rank by making a decision which of the eigenvalues are close enough to zero. Pseudo-inverse The pseudo inverse of a matrix A {\displaystyle A} is the unique matrix X = A + {\displaystyle X=A^{+}} for which A X {\displaystyle AX} and X A {\displaystyle XA} are symmetric and for which A X A = A , X A X ...

  6. 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.

  7. QR decomposition - Wikipedia

    en.wikipedia.org/wiki/QR_decomposition

    where R 1 is an n×n upper triangular matrix, 0 is an (m − n)×n zero matrix, Q 1 is m×n, Q 2 is m×(m − n), and Q 1 and Q 2 both have orthogonal columns. Golub & Van Loan (1996, §5.2) call Q 1 R 1 the thin QR factorization of A; Trefethen and Bau call this the reduced QR factorization. [1]

  8. 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 ...

  9. Rayleigh quotient - Wikipedia

    en.wikipedia.org/wiki/Rayleigh_quotient

    As stated in the introduction, for any vector x, one has (,) [,], where , are respectively the smallest and largest eigenvalues of .This is immediate after observing that the Rayleigh quotient is a weighted average of eigenvalues of M: (,) = = = = where (,) is the -th eigenpair after orthonormalization and = is the th coordinate of x in the eigenbasis.