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)