# Eigenvector

[ahy-guh n-vek-ter] /ˈaɪ gənˌvɛk tər/

**noun**, Mathematics.

1.

.

/ˈaɪɡənˌvɛktə/

**noun**

1.

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

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.

(1996-09-27)

Tagged: e

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