Eigenvector

linear-algebra

Definition

Eigenvector

An eigenvector of a linear mapping $f:V\to V$ is a non-zero vector $\mathbf{v}\in V$ that, when transformed by $f$, is scaled by a scalar factor $\lambda$ (the eigenvalue). Formally:

$f(\mathbf{v})=\lambda \mathbf{v},\mathbf{v}\ne 0$

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

Example

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

$$(\lambda I-A)\mathbf{v}=0$$

For a $2\times 2$ matrix with $\lambda =4$:

$$\left(\begin{array}{cc}4-a_{11}& -a_{12}\\ -a_{21}& 4-a_{22}\end{array}\right)\left(\begin{array}{c}{v}_1\\ v_2\end{array}\right)=\left(\begin{array}{c}0\\ 0\end{array}\right)$$

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