eigenvector

    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)