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 ?
It computes the eigenvectors by forming the product of all similarity transformations applied to A.
1], and the left eigenvectors are independent of b, so that all transition probability matrices are simultaneously diagonalized:
VD] = eig (L); L_eig_vec = [];% feature vector for i = 1 : size(V, 2)% for every f eature vector if (D (i, i) > 1)% the feature value is greater than 1 L_eig_vec = [L_eig_vec V (:, i)]; % concentrated corresponding eigenvectors end end Eigenfaces = A x L_eig_vec; Example function clear all clc close all Train Database Path = uigetdir ('D: \pca algorithm is used for the face recognition \PCA_basedFace Recognition System', .
Eigenvectors may show which variables are involved if there are large "loads" (values) for several variables for principal components of low eigenvalue (Figure 5, Principal Component 7 has large loads on X6 and XT).
0] = 0 into the fundamental eigenvector of the Wave Equation: [R.
Equation (6) shows, that the weight score of an element in any level can be computed by calculating the normalized eigenvector of the related matrix (based on the determination of pair wise comparison eigenvectors) and then multiple the result with the respective normalized weight-coordinate from the previous level.
The data matrix was transformed by the DCENTER algorithm using distances squared and eigenvectors and eigenvalues were calculated with NTSYS-pc verson 2.
In order to estimate the cost as a function of the original variables, the eigenvectors of the correlation matrix are multiplied in MVLR coefficients (B coefficients).
We change the number of eigenvectors from 2 to 50 with step 2 on the ORL, YALE face databases and the number of eigenvectors from 5 to 150 with step 5 on the AR face database, respectively.
The table of the eigenvectors of the main criteria indicates that financial perspective is the most important criteria (0.
For instance, in her linear algebra course she has used an article about eigenvectors to demonstrate viscerally an application of a mathematical concept—namely the math behind the algorithm that Google uses to put the most useful search results first.