A nonzero vector v such that Av = λv for some scalar λ (the corresponding eigenvalue); eigenvectors span eigenspaces and are defined up to nonzero scalar multiples.

Principle

Eigenvectors are the concrete carriers of invariant directions: the eigenspace for λ is ker(A−λI), a linear subspace whose dimension equals the geometric multiplicity of λ; they form a basis when the operator is diagonalizable.

Demonstration

For diag(5,3,3) in R^3, any nonzero scalar multiple of e1=(1,0,0) is an eigenvector for eigenvalue 5; the eigenspace for eigenvalue 3 is the span of e2 and e3 (a two-dimensional eigenspace) containing infinitely many eigenvectors.

Misapplication

Treating the zero vector as an eigenvector or assuming eigenvectors corresponding to the same eigenvalue are unique are common mistakes; also, assuming eigenvectors for distinct eigenvalues are always orthogonal is false unless the operator has special structure (e.g., normal).

Consequence

A complete set of linearly independent eigenvectors yields diagonalization and simplifies computations (powers, exponentials) and modal analysis; partial eigenvector information gives invariant subspaces used in reduction and perturbation theory.

Reversal

Generalized eigenvectors arise when (A−λI) is not diagonalizable: they satisfy (A−λI)^k v = 0 for some k>1 and extend eigenvectors to chains that capture nilpotent action, contrasting with ordinary eigenvectors which satisfy k=1.

Boundary

Eigenvectors are defined relative to a particular eigenvalue; the zero vector is excluded. Over fields that are not algebraically closed, some eigenvectors may only exist after extending scalars. Numerical approximations may produce near-eigenvectors not exactly invariant.

Semantic Tension

There is tension between eigenvectors and singular vectors: singular vectors pertain to A*A and optimal amplification in norms, while eigenvectors pertain to invariant directions of A itself; conflating them mixes distinct geometric information.

Synthesis

An eigenvector is a nonzero vector that a linear operator simply scales; collections of eigenvectors form eigenspaces that are the basic invariant building blocks for decomposing and understanding linear operators.