A nonzero vector in a vector space that an associated linear operator maps to a scalar multiple of itself, so that the operator acts by scaling (not rotating) on that one-dimensional direction.

Principle

An invariant direction of a linear transformation: applying the operator does not change the vector's direction, only its magnitude by a scalar factor that characterizes the action on that direction.

Demonstration

For a diagonal matrix with diagonal entries 2 and 3, the standard basis vectors are each mapped to scaled copies (by factors 2 and 3 respectively), so those basis vectors are eigenvectors of the matrix and span invariant coordinate directions.

Misapplication

Confusing an eigenvector with a vector in the kernel (mapped to zero) or treating every vector as decomposable into independent eigenvectors when the operator lacks a complete set of linearly independent invariant directions (defective operators).

Consequence

When a linear operator admits a basis of such invariant vectors, the operator is diagonalizable and many computations (powers, exponentials, spectral decompositions) reduce to scalar operations on those directions.

Reversal

Vectors that are not invariant directions: general vectors are typically mapped to different directions by the operator, and their images require decomposition into invariant subspaces to analyze behavior.

Boundary

The concept applies to linear maps on vector spaces (finite or infinite dimensional) over fields; it does not directly apply to nonlinear maps unless considering linearization about a point or generalized invariant structures like invariant subspaces without associated pure scaling.

Semantic Tension

Close to singular vectors (from the singular value decomposition): singular vectors are orthogonal directions that diagonalize the quadratic form A^T A and characterize input/output norms, whereas eigenvectors diagonalize the operator itself and depend on its potentially non-normal character.

Synthesis

An eigenvector is a nonzero vector that defines an invariant one-dimensional direction of a linear operator, on which the operator acts purely by multiplication by a scalar factor; collections of these directions organize the operator's reducible action.