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  2. Eigenvalues and eigenvectors of the second derivative

    en.wikipedia.org/wiki/Eigenvalues_and...

    These formulas are used to derive the expressions for eigenfunctions of Laplacian in case of separation of variables, as well as to find eigenvalues and eigenvectors of multidimensional discrete Laplacian on a regular grid, which is presented as a Kronecker sum of discrete Laplacians in one-dimension.

  3. Eigenvalues and eigenvectors - Wikipedia

    en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors

    where , the Hamiltonian, is a second-order differential operator and , the wavefunction, is one of its eigenfunctions corresponding to the eigenvalue , interpreted as its energy. However, in the case where one is interested only in the bound state solutions of the Schrödinger equation, one looks for ψ E {\displaystyle \psi _{E}} within the ...

  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. Eigenfunction - Wikipedia

    en.wikipedia.org/wiki/Eigenfunction

    Each value of λ corresponds to one or more eigenfunctions. If multiple linearly independent eigenfunctions have the same eigenvalue, the eigenvalue is said to be degenerate and the maximum number of linearly independent eigenfunctions associated with the same eigenvalue is the eigenvalue's degree of degeneracy or geometric multiplicity. [4] [5]

  6. Legendre polynomials - Wikipedia

    en.wikipedia.org/wiki/Legendre_polynomials

    From this standpoint, the eigenfunctions of the angular part of the Laplacian operator are the spherical harmonics, of which the Legendre polynomials are (up to a multiplicative constant) the subset that is left invariant by rotations about the polar axis.

  7. Self-adjoint operator - Wikipedia

    en.wikipedia.org/wiki/Self-adjoint_operator

    If we use the third choice of domain (with periodic boundary conditions), we can find an orthonormal basis of eigenvectors for A, the functions ():=. Thus, in this case finding a domain such that A is self-adjoint is a compromise: the domain has to be small enough so that A is symmetric, but large enough so that D ( A ∗ ) = D ( A ...

  8. 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.

  9. Spectral theory of ordinary differential equations - Wikipedia

    en.wikipedia.org/wiki/Spectral_theory_of...

    Indeed, the uniqueness of the fundamental eigenfunctions implies that its zeros are precisely the eigenvalues of D and that each non-zero eigenspace is one-dimensional. In particular there are at most countably many eigenvalues of D and, if there are infinitely many, they must tend to infinity.

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