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 Fourieranalysis 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 linearmap.

(1996-09-27)