Eigenvector

1 definition found

From The Free On-line Dictionary of Computing (27 SEP 03) [foldoc]:

  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)