**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 \(\lambda_2 = 3\).