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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. 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.

  4. 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.

  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. 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.

  7. Diagonalizable matrix - Wikipedia

    en.wikipedia.org/wiki/Diagonalizable_matrix

    The fundamental fact about diagonalizable maps and matrices is expressed by the following: An matrix over a field is diagonalizable if and only if the sum of the dimensions of its eigenspaces is equal to , which is the case if and only if there exists a basis of consisting of eigenvectors of .

  8. Moral Injury: The Grunts - The ... - The Huffington Post

    projects.huffingtonpost.com/moral-injury/the-grunts

    Most people enter military service “with the fundamental sense that they are good people and that they are doing this for good purposes, on the side of freedom and country and God,” said Dr. Wayne Jonas, a military physician for 24 years and president and CEO of the Samueli Institute, a non-profit health research organization.

  9. 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.