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  2. Eigenvalues and eigenvectors - Wikipedia

    en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors

    Moreover, if the entire vector space V can be spanned by the eigenvectors of T, or equivalently if the direct sum of the eigenspaces associated with all the eigenvalues of T is the entire vector space V, then a basis of V called an eigenbasis can be formed from linearly independent eigenvectors of T. When T admits an eigenbasis, T is ...

  3. 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.

  4. QR algorithm - Wikipedia

    en.wikipedia.org/wiki/QR_algorithm

    In the case when the matrix is depicted as a near-circle, the matrix can be replaced with one whose depiction is a perfect circle. In that case, the matrix is a multiple of the identity matrix, and its eigendecomposition is immediate. Be aware though that the resulting eigenbasis can be quite far from the original eigenbasis.

  5. Complete set of commuting observables - Wikipedia

    en.wikipedia.org/wiki/Complete_set_of_commuting...

    Proof that a common eigenbasis implies commutation. Let {| } be a set of orthonormal states (i.e., | =,) that form a complete eigenbasis for each of the two compatible observables and represented by the self-adjoint operators ^ and ^ with corresponding (real-valued) eigenvalues {} and {}, respectively.

  6. Position operator - Wikipedia

    en.wikipedia.org/wiki/Position_operator

    In literature, more or less explicitly, we find essentially three main directions to address this issue. The position operator is defined on the subspace D X {\displaystyle D_{X}} of L 2 {\displaystyle L^{2}} formed by those equivalence classes ψ {\displaystyle \psi } whose product by the embedding x {\displaystyle \mathrm {x} } lives in the ...

  7. 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 ...

  8. 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 ...

  9. Singular value decomposition - Wikipedia

    en.wikipedia.org/wiki/Singular_value_decomposition

    The geometric content of the SVD theorem can thus be summarized as follows: for every linear map ⁠: ⁠ one can find orthonormal bases of ⁠ ⁠ and ⁠ ⁠ such that ⁠ ⁠ maps the ⁠ ⁠-th basis vector of ⁠ ⁠ to a non-negative multiple of the ⁠ ⁠-th basis vector of ⁠, ⁠ and sends the leftover basis vectors to zero.