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  2. Spectrum of a matrix - Wikipedia

    en.wikipedia.org/wiki/Spectrum_of_a_matrix

    Thus the elements of the spectrum are precisely the eigenvalues of T, and the multiplicity of an eigenvalue λ in the spectrum equals the dimension of the generalized eigenspace of T for λ (also called the algebraic multiplicity of λ). Now, fix a basis B of V over K and suppose M ∈ Mat K (V) is a matrix.

  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. 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 .

  5. 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 ...

  6. 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.

  7. Spectrum (functional analysis) - Wikipedia

    en.wikipedia.org/wiki/Spectrum_(functional_analysis)

    If is an eigenvalue of , then the operator is not one-to-one, and therefore its inverse () is not defined. However, the converse statement is not true: the operator may not have an inverse, even if is not an eigenvalue.

  8. Decomposition of spectrum (functional analysis) - Wikipedia

    en.wikipedia.org/wiki/Decomposition_of_spectrum...

    Alternatively, if it is insisted that the notion of eigenvectors and eigenvalues survive the passage to the rigorous, one can consider operators on rigged Hilbert spaces. [8] An example of an observable whose spectrum is purely absolutely continuous is the position operator of a free particle moving on

  9. 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.