##  [Eigenvector](/eigenvector-0) 

 Definition

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

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

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

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

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

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

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

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

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.