eigenvector

ei·gen·vec·tor

E5058990 (ī′gən-vĕk′tər)n. A vector whose direction is unchanged by a given transformation and whose magnitude is changed by a factor corresponding to that vector's eigenvalue. In quantum mechanics, the transformations involved are operators corresponding to a physical system's observables. The eigenvectors correspond to possible states of the system, and the eigenvalues to possible observed values.

[Partial translation of German Eigenvektor : eigen-, characteristic; see eigenvalue + Vektor, vector.]

eigenvector

(ˈaɪɡənˌvɛktə) n (Mathematics) maths physics a vector x satisfying an equation Ax = λx, where A is a square matrix and λ is a constant

Translations

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 particularlinear transformation, produces a scalar multiple of theoriginal vector. The scalar in question is called theeigenvalue corresponding to this eigenvector.

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

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

An eigenvector is a (representative member of a) fixed pointof the map on the projective plane induced by a linear map.

单词	eigenvector
释义	eigenvector ei·gen·vec·tor E5058990 (ī′gən-vĕk′tər)n. A vector whose direction is unchanged by a given transformation and whose magnitude is changed by a factor corresponding to that vector's eigenvalue. In quantum mechanics, the transformations involved are operators corresponding to a physical system's observables. The eigenvectors correspond to possible states of the system, and the eigenvalues to possible observed values. [Partial translation of German Eigenvektor : eigen-, characteristic; see eigenvalue + Vektor, vector.] eigenvector (ˈaɪɡənˌvɛktə) n (Mathematics) maths physics a vector x satisfying an equation Ax = λx, where A is a square matrix and λ is a constant Translations eigenvector 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 particularlinear transformation, produces a scalar multiple of theoriginal vector. The scalar in question is called theeigenvalue corresponding to this eigenvector. It should be noted that "vector" here means "element of avector space" which can include many mathematical entities.Ordinary vectors are elements of a vector space, andmultiplication by a matrix is a linear transformation onthem; smooth functions "are vectors", and many partialdifferential operators are linear transformations on the spaceof such functions; quantum-mechanical states "are vectors",and observables are linear transformations on the statespace. An important theorem says, roughly, that certain lineartransformations have enough eigenvectors that they form abasis of the whole vector states. This is why Fourier analysis works, and why in quantum mechanics every state is asuperposition of eigenstates of observables. An eigenvector is a (representative member of a) fixed pointof the map on the projective plane induced by a linear map. AcronymsSeeEVI
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