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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. 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.

  5. 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.

  6. Singular value decomposition - Wikipedia

    en.wikipedia.org/wiki/Singular_value_decomposition

    Yet another usage is latent semantic indexing in natural-language text processing. In general numerical computation involving linear or linearized systems, there is a universal constant that characterizes the regularity or singularity of a problem, which is the system's "condition number" κ := σ max / σ min {\displaystyle \kappa :=\sigma ...

  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. Frobenius covariant - Wikipedia

    en.wikipedia.org/wiki/Frobenius_covariant

    The Frobenius covariants of a matrix A can be obtained from any eigendecomposition A = SDS −1, where S is non-singular and D is diagonal with D i,i = λ i. If A has no multiple eigenvalues, then let c i be the i th right eigenvector of A, that is, the i th column of S; and let r i be the i th left eigenvector of A, namely the i th row of S ...

  9. Eigenfunction - Wikipedia

    en.wikipedia.org/wiki/Eigenfunction

    This solution of the vibrating drum problem is, at any point in time, an eigenfunction of the Laplace operator on a disk.. In mathematics, an eigenfunction of a linear operator D defined on some function space is any non-zero function in that space that, when acted upon by D, is only multiplied by some scaling factor called an eigenvalue.