<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.
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