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 themsmooth functions "are vectors", and many partial differential operators are linear transformations on the space of such functionsquantum-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)