Definition of eigenvector/characteristic vector of a linear transformation
Created almost 9 years ago, updated 5 months ago
ID24043
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.
**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.