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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. Riesz projector - Wikipedia

    en.wikipedia.org/wiki/Riesz_projector

    In mathematics, or more specifically in spectral theory, the Riesz projector is the projector onto the eigenspace corresponding to a particular eigenvalue of an operator (or, more generally, a projector onto an invariant subspace corresponding to an isolated part of the spectrum).

  4. Dirichlet eigenvalue - Wikipedia

    en.wikipedia.org/wiki/Dirichlet_eigenvalue

    This operator is invertible, and its inverse is compact and self-adjoint so that the usual spectral theorem can be applied to obtain the eigenspaces of Δ and the reciprocals 1/λ of its eigenvalues. One of the primary tools in the study of the Dirichlet eigenvalues is the max-min principle: the first eigenvalue λ 1 minimizes the Dirichlet ...

  5. Eigenvalues and eigenvectors - Wikipedia

    en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors

    The set of all eigenvectors of T corresponding to the same eigenvalue, together with the zero vector, is called an eigenspace, or the characteristic space of T associated with that eigenvalue. [ 9 ] If a set of eigenvectors of T forms a basis of the domain of T , then this basis is called an eigenbasis .

  6. Characteristic polynomial - Wikipedia

    en.wikipedia.org/wiki/Characteristic_polynomial

    In linear algebra, eigenvalues and eigenvectors play a fundamental role, since, given a linear transformation, an eigenvector is a vector whose direction is not changed by the transformation, and the corresponding eigenvalue is the measure of the resulting change of magnitude of the vector.

  7. Jordan normal form - Wikipedia

    en.wikipedia.org/wiki/Jordan_normal_form

    This shows that the eigenvalues are 1, 2, 4 and 4, according to algebraic multiplicity. The eigenspace corresponding to the eigenvalue 1 can be found by solving the equation Av = λv. It is spanned by the column vector v = (−1, 1, 0, 0) T. Similarly, the eigenspace corresponding to the eigenvalue 2 is spanned by w = (1, −1, 0, 1) T.

  8. Spectral theory of compact operators - Wikipedia

    en.wikipedia.org/wiki/Spectral_theory_of_compact...

    Let X(λ) = E(λ)X if λ is a non-zero eigenvalue. Thus X(λ) is a finite-dimensional invariant subspace, the generalised eigenspace of λ. Let X(0) be the intersection of the kernels of the E(λ). Thus X(0) is a closed subspace invariant under C and the restriction of C to X(0) is a compact operator with spectrum {0}.

  9. Spread of a matrix - Wikipedia

    en.wikipedia.org/wiki/Spread_of_a_matrix

    In both cases, all eigenvalues are equal, so no two eigenvalues can be at nonzero distance from each other. For a projection , the only eigenvalues are zero and one. A projection matrix therefore has a spread that is either 0 {\displaystyle 0} (if all eigenvalues are equal) or 1 {\displaystyle 1} (if there are two different eigenvalues).