Definition, Meaning & Synonyms | English word EIGENVECTOR

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Definitions of EIGENVECTOR

1. (linear algebra) A vector that is not rotated under a particular linear transformation; a left or right eigenvector depending on context; (more formally) given a linear transformation A, a vector x such that Ax=λx [or xA=λx] for some scalar λ (called the eigenvalue).
2. (physics, engineering) A right eigenvector; given a matrix A, the eigenvector of the transformation "left-side multiplication by A."

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Examples of Using EIGENVECTOR in a Sentence

• The sizing and layout of the autonomous systems was calculated based on their eigenvector centrality, which is a measure of how central to the network each autonomous system 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.
• The adjacency matrix of an undirected simple graph is symmetric, and therefore has a complete set of real eigenvalues and an orthogonal eigenvector basis.
• It is also used in eigenvalue algorithms (such as Rayleigh quotient iteration) to obtain an eigenvalue approximation from an eigenvector approximation.
• This can be proven by using induction on the fact that A has an eigenvector, by taking the quotient space by the eigenvector and inducting to show that A stabilizes a flag, and is thus triangularizable with respect to a basis for that flag.
• As any rotation matrix has a single real eigenvalue, which is equal to +1, the corresponding eigenvector defines the rotation axis.
• The above-mentioned Euler vector is the eigenvector of a rotation matrix (a rotation matrix has a unique real eigenvalue).
• The Rayleigh quotient iteration algorithm converges cubically for Hermitian or symmetric matrices, given an initial vector that is sufficiently close to an eigenvector of the matrix that is being analyzed.
• These algorithms generally aim to determine the ground-state eigenvector and eigenvalue of a Hermitian operator.
• For example, eigenvector centrality uses the eigenvectors of the adjacency matrix corresponding to a network, to determine nodes that tend to be frequently visited.
• The transformation is also known as Hotelling transform and eigenvector transform, and is closely related to principal component analysis (PCA) technique widely used in image processing and in data analysis in many fields.
• The sparsest cut of a graph can be approximated through the Fiedler vector — the eigenvector corresponding to the second smallest eigenvalue of the graph Laplacian — as established by Cheeger's inequality.
• Restricting consideration to this group allows for taxonomy which places many centralities on a spectrum from those concerned with walks of length one (degree centrality) to infinite walks (eigenvector centrality).
• In using these criteria, one must be careful to ensure that the largest eigenvalue belongs to a timelike eigenvector, since there are Lorentzian manifolds, satisfying this eigenvalue criterion, in which the large eigenvalue belongs to a spacelike eigenvector, and these cannot represent radiation fluids.
• The Lanczos algorithm is most often brought up in the context of finding the eigenvalues and eigenvectors of a matrix, but whereas an ordinary diagonalization of a matrix would make eigenvectors and eigenvalues apparent from inspection, the same is not true for the tridiagonalization performed by the Lanczos algorithm; nontrivial additional steps are needed to compute even a single eigenvalue or eigenvector.
• A Metzler matrix has an eigenvector in the nonnegative orthant because of the corresponding property for nonnegative matrices.
• Using the fact that it is an eigenvector of the Casimir operator and has an eigenvector for H, it follows easily that any irreducible admissible representation is a subrepresentation of a parabolically induced representation.
• The PageRank values are the entries of the dominant right eigenvector of the modified adjacency matrix rescaled so that each column adds up to one.
• It can be shown that a Bethe vector is an eigenvector of the Gaudin Hamiltonians if the set of equations.

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