Eigenvector

linear-algebra

Definition

Eigenvector

Let be a linear operator on . An element is an eigenvector of if:

where is called the eigenvalue of the eigenvector .

Geometric Interpretation

Geometrically, an eigenvector points in a direction that remains invariant under the linear operation, with its magnitude being adjusted by .

Example

To find the eigenvectors of an eigenvalue for a matrix , solve the homogeneous linear system:

For a matrix with :

;
The solution space (the eigenspace) consists of all vectors that satisfy this equality.