AThemATics> 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)