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 ?
To visualize the perturbed unstable and stable manifolds, one should find the exact location of the perturbed hyperbolic fixed points and the associated directions of eigenvectors.
IBM's solution, named CLEVER (client-side eigenvector enhanced retrieval), is to compile automatically a resource list on any widely present topic on the Web.
If the alpha is zero, pagerank centrality is equal to eigenvector centrality (Rubinov and Sporns, 2010).
Essentially, this strategy consists of applying the MINRES method to a projected linear system where the projection is built from approximate eigenvectors for the first matrix of the sequence.
One of the most well known methods to do that is Principal Component Analysis PCA [13] where projection of the initial data is performed into reduced space by a given number of the highest eigenvectors.
is the objective function of PCA , w , which is the solution of the objective function, is the eigenvector associated with the largest eigenvalue of the covariance matrix [S.
i] in terms of eigenvectors of the above eigenvalue problem and observations s(t).
2]) for an eigenvector collinear to v, the two other eigenvalues being equal to zero.
With PCA, 65% of the variability in the monthly oceanographic (O) data sets was explained by the first eigenvector ([PC1.
Denoting K([epsilon]) = D([member of])G([member of])the new matrix, K([epsilon]) is primitive for a > 0 small enough, thus, it has a unique and positive leading eigenvalue [lambda]([epsilon]) associated with a positive right eigenvector v([epsilon]) and a positive left eigenvector w([epsilon]).
The priorities are derived from the elements of the eigenvector with the largest eigenvalue.