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  2. Eigenvalues and eigenvectors - Wikipedia

    en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors

    The eigenvectors and eigenvalues of a linear transformation serve ... The classical method is to first find the eigenvalues, and then calculate the eigenvectors for ...

  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

    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.

  5. 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.

  6. 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).

  7. Principal component analysis - Wikipedia

    en.wikipedia.org/wiki/Principal_component_analysis

    The eigenvalues and eigenvectors are ordered and paired. The jth eigenvalue corresponds to the jth eigenvector. Matrix V denotes the matrix of right eigenvectors (as opposed to left eigenvectors). In general, the matrix of right eigenvectors need not be the (conjugate) transpose of the matrix of left eigenvectors. Rearrange the eigenvectors and ...

  8. 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.

  9. Lanczos algorithm - Wikipedia

    en.wikipedia.org/wiki/Lanczos_algorithm

    The Lanczos algorithm is most often brought up in the context of finding the eigenvalues and eigenvectors of a matrix, but whereas an ordinary diagonalization of a matrix would make eigenvectors and eigenvalues apparent from inspection, the same is not true for the tridiagonalization performed by the Lanczos algorithm; nontrivial additional steps are needed to compute even a single eigenvalue ...