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,..., x_n of the eigenvectors of a transformation of n-dimensional space with the matrix ║a_ik║ 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.

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References in periodicals archive ?

In order to maximize the ergodic SNR, the optimized precoded signals to user A should be along the dominant eigenvector of [R.sub.A] namely [v.sub.A] = [u.sub.max]([R.sub.A]).

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In this case, only when the eigenvalue [[lambda].sub.i] of the G takes the maximum, the corresponding eigenvector [u.sub.i] is the maximum value.

Image Recognition Based on Two-Dimensional Principal Component Analysis Combining with Wavelet Theory and Frame Theory

It shows two ways for how to prove the well-known statement regarding the set of eigenvalues of Aw = [lambda]w with a consistent matrix A of type n x n (n [greater than or equal to] 2), and it demonstrates how to find the corresponding eigenvector components presented as some probabilities of events for maximal eigenvalue [lambda] = [[lambda].sub.max] = n.

Two Proofs and One Algorithm Related to the Analytic Hierarchy Process

Patient 1664 was favoured (based on degree, closeness, betweenness, and eigenvector network centrality metrics) as the most important in the transmission network by having the highest number of secondary cases.

Network Analysis of MERS Coronavirus within Households, Communities, and Hospitals to Identify Most Centralized and Super-Spreading in the Arabian Peninsula, 2012 to 2016

Additionally, we can factor [R.sub.c] to be [R.sub.c] = [U.sub.0][[lambda].sub.c][U.sup.T.sub.0], where [U.sub.0] [member of] [R.sup.nxn] represents a matrix with eigenvectors in a row, while [[LAMBDA].sub.c] represents the diagonal matrix of eigenvalues classified in declining order.

Transfer Kernel Common Spatial Patterns for Motor Imagery Brain-Computer Interface Classification

When [S.sub.WT] is nonsingular, the basis vectors [psi] in (3) correspond to the first M([less than or equal to]N) eigenvectors with the largest eigenvalues of [S.sup.-1.sub.WT][S.sub.BT].

Identification of Cerebral Artery Stenosis Using Bilateral Photoplethysmography

In |t> representation, the operator of energy becomes i[??]([partial derivative]/[partial derivative]t), while its eigenvectors |E> become [e.sup.(1/i[??])Et], for every E [member of] R.

Operator of Time and Generalized Schrodinger Equation

Compute the cumulative content for each Eigenvector

An intelligent face recognition system using eigen feature approach for crime investigation

and the optimal a's are also the maximum eigenvectors of eigen-problem:

Image Retrieval Using Deep Convolutional Neural Networks and Regularized Locality Preserving Indexing Strategy

Caption: FIGURE 3: Color eigenvector: V first-order moment.

Road Surface State Recognition Based on SVM Optimization and Image Segmentation Processing

In order to speed up the k-means clustering on the embedded eigenvector matrix, we sample row vectors of eigenvectors matrix randomly and get k centers through k-means clustering over the selected row vectors.

Fast Constrained Spectral Clustering and Cluster Ensemble with Random Projection

The variance of the second to the seventh eigenvector of empirical orthogonal function account for 2.04%, 1.09%, 0.97%, 0.44%, 0.41%, and 0.28%, respectively.

Detection of Decreasing Vegetation Cover Based on Empirical Orthogonal Function and Temporal Unmixing Analysis