**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.