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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.
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.
The k-th principal component of a data vector x (i) can therefore be given as a score t k(i) = x (i) ⋅ w (k) in the transformed coordinates, or as the corresponding vector in the space of the original variables, {x (i) ⋅ w (k)} w (k), where w (k) is the kth eigenvector of X T X. The full principal components decomposition of X can therefore ...
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.
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.
either the eigenvector r is a null vector (i.e. η(r,r) = 0), or the eigenvalue λ is zero, or both. A 1-dimensional subspace generated by a null eigenvector is called a principal null direction of the bivector. The classification theorem characterizes the possible principal null directions of a bivector.
If A is a N × N matrix, X a non-zero vector, and λ is a scalar, such that =, then the scalar λ is said to be an eigenvalue of A and the vector X is said to be the eigenvector corresponding to λ. Together with the zero vector, the set of all eigenvectors corresponding to a given eigenvalue λ form a subspace of C n , which is called the ...
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.