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The th principal eigenvector of a graph is defined as either the eigenvector corresponding to the th largest or th smallest eigenvalue of the Laplacian. The first principal eigenvector of the graph is also referred to merely as the principal eigenvector.
In mathematics, power iteration (also known as the power method) is an eigenvalue algorithm: given a diagonalizable matrix, the algorithm will produce a number , which is the greatest (in absolute value) eigenvalue of , and a nonzero vector , which is a corresponding eigenvector of , that is, =.
In general, an eigenvector of a linear operator D defined on some vector space is a nonzero vector in the domain of D that, when D acts upon it, is simply scaled by some scalar value called an eigenvalue. In the special case where D is defined on a function space, the eigenvectors are referred to as eigenfunctions.
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 ...
Hence PageRank is the principal eigenvector of ^. A fast and easy way to compute this is using the power method : starting with an arbitrary vector x ( 0 ) {\displaystyle x(0)} , the operator M ^ {\displaystyle {\widehat {\mathcal {M}}}} is applied in succession, i.e.,
Let = be an positive matrix: > for ,.Then the following statements hold. There is a positive real number r, called the Perron root or the Perron–Frobenius eigenvalue (also called the leading eigenvalue, principal eigenvalue or dominant eigenvalue), such that r is an eigenvalue of A and any other eigenvalue λ (possibly complex) in absolute value is strictly smaller than r, |λ| < r.
Calculate the eigenvectors and eigenvalues of the covariance matrix S. Each eigenvector has the same dimensionality (number of components) as the original images, and thus can itself be seen as an image. The eigenvectors of this covariance matrix are therefore called eigenfaces. They are the directions in which the images differ from the mean ...
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.