Definition of eigenvalue/characteristic value of a linear transformation

Definition: Let (T: V o V) be a linear transformation on a vector space (V). A scalar (\lambda) (from the underlying field) is an eigenvalue (or characteristic value) of (T) if there exists a nonzero vector (\mathbf{v} \in V) such that:
[T(\mathbf{v}) = \lambda \mathbf{v}]

The eigenvalue (\lambda) is a scalar by which the transformation scales some nonzero vector.

Intuition: An eigenvalue is a factor by which the transformation stretches or compresses vectors along certain directions. If (\lambda > 1), the vector is stretched; if (0 < \lambda < 1), it is compressed; if (\lambda < 0), the direction is reversed.

Example: For the transformation (T(x,y) = (2x, 3y)), the eigenvalues are (\lambda_1 = 2) and (\lambda_2 = 3).