eigenvector

eigenvector

   <mathematics> 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)