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  2. 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 ...

  3. Rotation matrix - Wikipedia

    en.wikipedia.org/wiki/Rotation_matrix

    For example, in 2-space n = 2, a rotation by angle θ has eigenvalues λ = e iθ and λ = e −iθ, so there is no axis of rotation except when θ = 0, the case of the null rotation. In 3-space n = 3, the axis of a non-null proper rotation is always a unique line, and a rotation around this axis by angle θ has eigenvalues λ = 1, e iθ, e −iθ.

  4. Eigenvalues and eigenvectors - Wikipedia

    en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors

    Eigenvalues and eigenvectors give rise to many closely related mathematical concepts, and the prefix eigen-is applied liberally when naming them: The set of all eigenvectors of a linear transformation, each paired with its corresponding eigenvalue, is called the eigensystem of that transformation. [7] [8]

  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. Transformation matrix - Wikipedia

    en.wikipedia.org/wiki/Transformation_matrix

    The surviving diagonal elements, ,, are known as eigenvalues and designated with in the defining equation, which reduces to =. The resulting equation is known as eigenvalue equation . [ 5 ] The eigenvectors and eigenvalues are derived from it via the characteristic polynomial .

  8. Schur decomposition - Wikipedia

    en.wikipedia.org/wiki/Schur_decomposition

    The generalized eigenvalues that solve the generalized eigenvalue problem = (where x is an unknown nonzero vector) can be calculated as the ratio of the diagonal elements of S to those of T. That is, using subscripts to denote matrix elements, the i th generalized eigenvalue λ i {\displaystyle \lambda _{i}} satisfies λ i = S i i / T i i ...

  9. Sylvester's formula - Wikipedia

    en.wikipedia.org/wiki/Sylvester's_formula

    If a matrix A is both Hermitian and unitary, then it can only have eigenvalues of , and therefore = +, where + is the projector onto the subspace with eigenvalue +1, and is the projector onto the subspace with eigenvalue ; By the completeness of the eigenbasis, + + =.