Definition of eigenvalue/characteristic value of a linear transformation
Created almost 9 years ago, updated 5 months ago
ID24042
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).
**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 \(\lambda_2 = 3\).