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  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

    On the other hand, by definition, any nonzero vector that satisfies this condition is an eigenvector of A associated with λ. So, the set E is the union of the zero vector with the set of all eigenvectors of A associated with λ, and E equals the nullspace of (A − λI). E is called the eigenspace or characteristic space of A associated with λ.

  4. Matrix decomposition - Wikipedia

    en.wikipedia.org/wiki/Matrix_decomposition

    For a normal matrix A (and only for a normal matrix), the eigenvectors can also be made orthonormal (=) and the eigendecomposition reads as =. In particular all unitary , Hermitian , or skew-Hermitian (in the real-valued case, all orthogonal , symmetric , or skew-symmetric , respectively) matrices are normal and therefore possess this property.

  5. Singular value decomposition - Wikipedia

    en.wikipedia.org/wiki/Singular_value_decomposition

    ⁠ By the definition of a unitary matrix, the same is true for their conjugate transposes ⁠ ⁠ and ⁠, ⁠ except the geometric interpretation of the singular values as stretches is lost. In short, the columns of ⁠ U , {\displaystyle \mathbf {U} ,} ⁠ ⁠ U ∗ , {\displaystyle \mathbf {U} ^{*},} ⁠ ⁠ V , {\displaystyle \mathbf {V ...

  6. Spectrum of a matrix - Wikipedia

    en.wikipedia.org/wiki/Spectrum_of_a_matrix

    The eigendecomposition (or spectral decomposition) of a diagonalizable matrix is a decomposition of a diagonalizable matrix into a specific canonical form whereby the matrix is represented in terms of its eigenvalues and eigenvectors. The spectral radius of a square matrix is the largest absolute value of its eigenvalues.

  7. Eigendecomposition - Wikipedia

    en.wikipedia.org/?title=Eigendecomposition&...

    This page was last edited on 25 November 2009, at 17:24 (UTC).; Text is available under the Creative Commons Attribution-ShareAlike 4.0 License; additional terms may apply.

  8. 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.

  9. Nonlinear eigenproblem - Wikipedia

    en.wikipedia.org/wiki/Nonlinear_eigenproblem

    Definition: An eigenvalue is said to have algebraic multiplicity if is the smallest integer such that the th derivative of (()) with respect to , in is nonzero.