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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 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.
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 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 ...
Any eigenvector is a generalized eigenvector, and so each eigenspace is contained in the associated generalized eigenspace. This provides an easy proof that the geometric multiplicity is always less than or equal to the algebraic multiplicity. This usage should not be confused with the generalized eigenvalue problem described below.
In quantum mechanics, the eigenvalue of an observable is said to be a good quantum number if the observable is a constant of motion.In other words, the quantum number is good if the corresponding observable commutes with the Hamiltonian.
It is an eigenvector of the energy operator (instead of a quantum superposition of different energies). It is also called energy eigenvector, energy eigenstate, energy eigenfunction, or energy eigenket. It is very similar to the concept of atomic orbital and molecular orbital in chemistry, with some slight differences explained below.
As stated in the introduction, for any vector x, one has (,) [,], where , are respectively the smallest and largest eigenvalues of .This is immediate after observing that the Rayleigh quotient is a weighted average of eigenvalues of M: (,) = = = = where (,) is the -th eigenpair after orthonormalization and = is the th coordinate of x in the eigenbasis.