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  2. Eigenvalues and eigenvectors - Wikipedia

    en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors

    In spectral graph theory, an eigenvalue of a graph is defined as an eigenvalue of the graph's adjacency matrix, or (increasingly) of the graph's Laplacian matrix due to its discrete Laplace operator, which is either (sometimes called the combinatorial Laplacian) or / / (sometimes called the normalized Laplacian), where is a diagonal matrix with ...

  3. Eigenvalues and eigenvectors of the second derivative

    en.wikipedia.org/wiki/Eigenvalues_and...

    Notation: The index j represents the jth eigenvalue or eigenvector. The index i represents the ith component of an eigenvector. Both i and j go from 1 to n, where the matrix is size n x n. Eigenvectors are normalized. The eigenvalues are ordered in descending order.

  4. 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.

  5. Adjacency matrix - Wikipedia

    en.wikipedia.org/wiki/Adjacency_matrix

    The relationship between a graph and the eigenvalues and eigenvectors of its adjacency matrix is studied in spectral graph theory. The adjacency matrix of a graph should be distinguished from its incidence matrix , a different matrix representation whose elements indicate whether vertex–edge pairs are incident or not, and its degree matrix ...

  6. Eigenvector centrality - Wikipedia

    en.wikipedia.org/wiki/Eigenvector_centrality

    In graph theory, eigenvector centrality (also called eigencentrality or prestige score [1]) is a measure of the influence of a node in a connected network.Relative scores are assigned to all nodes in the network based on the concept that connections to high-scoring nodes contribute more to the score of the node in question than equal connections to low-scoring nodes.

  7. Spectral graph theory - Wikipedia

    en.wikipedia.org/wiki/Spectral_graph_theory

    In mathematics, spectral graph theory is the study of the properties of a graph in relationship to the characteristic polynomial, eigenvalues, and eigenvectors of matrices associated with the graph, such as its adjacency matrix or Laplacian matrix.

  8. 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 ...

  9. Phase plane - Wikipedia

    en.wikipedia.org/wiki/Phase_plane

    where λ 1 and λ 2 are the eigenvalues, and (k 1, k 2), (k 3, k 4) are the basic eigenvectors. The constants c 1 and c 2 account for the nonuniqueness of eigenvectors and are not solvable unless an initial condition is given for the system. The above determinant leads to the characteristic polynomial: