enow.com Web Search

Search results

  1. Results from the WOW.Com Content Network
  2. 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.

  3. Eigenvalues and eigenvectors - Wikipedia

    en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors

    This equation is called the eigenvalue equation for T, and the scalar λ is the eigenvalue of T corresponding to the eigenvector v. T(v) is the result of applying the transformation T to the vector v, while λv is the product of the scalar λ with v. [37] [38]

  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. Singular value decomposition - Wikipedia

    en.wikipedia.org/wiki/Singular_value_decomposition

    ⁠ For every unit length eigenvector ⁠ ⁠ of ⁠ ⁠ its eigenvalue is ⁠ (), ⁠ so ⁠ ⁠ is the largest eigenvalue of ⁠. ⁠ The same calculation performed on the orthogonal complement of ⁠ u {\displaystyle \mathbf {u} } ⁠ gives the next largest eigenvalue and so on.

  6. Matrix decomposition - Wikipedia

    en.wikipedia.org/wiki/Matrix_decomposition

    Comment: in the complex QZ decomposition, the ratios of the diagonal elements of S to the corresponding diagonal elements of T, = /, are the generalized eigenvalues that solve the generalized eigenvalue problem = (where is an unknown scalar and v is an unknown nonzero vector).

  7. 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 ...

  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. Perron–Frobenius theorem - Wikipedia

    en.wikipedia.org/wiki/Perron–Frobenius_theorem

    Let = be an positive matrix: > for ,.Then the following statements hold. There is a positive real number r, called the Perron root or the Perron–Frobenius eigenvalue (also called the leading eigenvalue, principal eigenvalue or dominant eigenvalue), such that r is an eigenvalue of A and any other eigenvalue λ (possibly complex) in absolute value is strictly smaller than r, |λ| < r.