Eigenvector

linear-algebra

Definition

Eigenvector

Let $T:V\to V$ be a linear operator on $V$. An element $x\in V$ is an eigenvector of $T$ if:

$$x\ne 0\wedge \exists \lambda \in \mathbb{C}:Tx=\lambda x$$

where $\lambda$ is called the eigenvalue of the eigenvector $x$.

Geometric Interpretation

Geometrically, an eigenvector points in a direction that remains invariant under the linear operation, 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 $\lambda =4$:

$$\left(\begin{array}{cc}4-a_{11}& -a_{12}\\ -a_{21}& 4-a_{22}\end{array}\right)\left(\begin{array}{c}{v}_1{v}_2\end{array}\right)=\left(\begin{array}{c}00\end{array}\right)$$

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