eigenvector


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eigenvector

[′ī·gən‚vek·tər]
(mathematics)
A nonzero vector v whose direction is not changed by a given linear transformation T ; that is, T (v) = λ v for some scalar λ. Also known as characteristic vector.

Eigenvector

 

(or characteristic vector). An eigenvector of a linear transformation is a vector that does not change direction under the transformation and is simply multiplied by a scalar. For example, the eigenvectors of a transformation composed of rotations about some axis and of contraction toward the plane perpendicular to the axis are vectors directed along the axis.

The coordinates x1x2,..., xn of the eigenvectors of a transformation of n-dimensional space with the matrix ║aik║ satisfy the system of homogeneous linear equations

where λ is an eigenvalue of the matrix. If the matrix of a transformation is Hermitian, then the eigenvectors are mutually perpendicular. As a result of a Hermitian transformation, a sphere becomes an ellipsoid whose major axes are eigenvectors of the transformation.

eigenvector

(mathematics)
A vector which, when acted on by a particular linear transformation, produces a scalar multiple of the original vector. The scalar in question is called the eigenvalue corresponding to this eigenvector.

It should be noted that "vector" here means "element of a vector space" which can include many mathematical entities. Ordinary vectors are elements of a vector space, and multiplication by a matrix is a linear transformation on them; smooth functions "are vectors", and many partial differential operators are linear transformations on the space of such functions; quantum-mechanical states "are vectors", and observables are linear transformations on the state space.

An important theorem says, roughly, that certain linear transformations have enough eigenvectors that they form a basis of the whole vector states. This is why Fourier analysis works, and why in quantum mechanics every state is a superposition of eigenstates of observables.

An eigenvector is a (representative member of a) fixed point of the map on the projective plane induced by a linear map.
References in periodicals archive ?
where [mathematical expression not reproducible] is the measured eigenvector and [[phi].
Reanalysis methods estimate the eigenvalues and eigenvectors of a modified design without performing a full-order eigenvalue analysis.
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To demonstrate the accuracy after the eigenvector elimination of the three basis sets of Table 4, a unit excitation is applied on the outer (Point 1) and inner (Point 2) panels of the door (Figure 4) and the magnitude of the velocity frequency response is calculated using PROM.
M] is of rank one, its principal eigenvector is the solution of the original problem.
Then, the gKDR proposes the eigenvectors of M(p + q) x M(p + q) symmetric matrix:
In this paper we also give exploitation of the eigenvector [q.
Eryilmaz [6] studied q-Sturm-Liouville boundary value problem in the Hilbert space with a spectral parameter in the boundary condition and he proved theorems on the completeness of the system of eigenvalues and eigenvectors of operator by Pavlov's method.
The higher the eigenvector centrality score, the more important the journal.
SELECTED EIGENVALUES AND THEIR CORRESPONDING EIGENVECTOR (LIQUIDITY RATIOS, SMALLER-BETTER PRINCIPLE FOR 2011) eigenvalue 17.