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Definition of eigenvector/characteristic vector of a linear transformation

Created over 8 years ago, updated 10 days ago

Definition: Let (T: V o V) be a linear transformation and (\lambda) an eigenvalue of (T). A nonzero vector (\mathbf{v} \in V) is an eigenvector (or characteristic vector) of (T) corresponding to (\lambda) if:
[T(\mathbf{v}) = \lambda \mathbf{v}]

Eigenvectors are the special vectors whose direction is unchanged by the transformation; they are only scaled.

Example: For (T(x,y) = (2x, 3y)):

  • ((1,0)) is an eigenvector with eigenvalue (2) since (T(1,0) = (2,0) = 2(1,0))
  • ((0,1)) is an eigenvector with eigenvalue (3) since (T(0,1) = (0,3) = 3(0,1))

Important: The zero vector is never considered an eigenvector, even though (T(\mathbf{0}) = \lambda \mathbf{0}) holds trivially.