<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.
