Dictionary.net

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 Fourieranalysis 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 linearmap.