# Description

An **eigenvector** of a linear transformation is a **non-trivial** vector that is only **scaled** when a linear transformation is applied to it. This means that the direction of the eigenvector is preserved after the operation.

In mathematical notation: Let $$\vec{v}$$ be an eigenvector of the matrix $$\textbf{A}$$. Then $$\textbf{A}\vec{v} = \lambda\vec{v}$$ for a unique $$\lambda \in \mathbb{R}$$.

This equation says that the transformed vector $$\textbf{A}\vec{x}$$ is just a scaled version of the original vector $$\vec{v}$$.

We also define the eigenvector to be non-trivial because $$\textbf{A}\vec{0} = \vec{0}$$ for all matrices.

The **eigenvalue** associated with this eigenvector is the scalar quantity by which the eigenvector is scaled. In the above equation, $$\lambda$$ would the eigenvalue associated with the vector $$\vec{v}$$. The eigenvalue associated with an eigenvector can be 0.

The **eigenspace** of an eigenvalue is the span of all eigenvectors with this eigenvalue because any linear combination of the eigenvectors is still an eigenvector with the **same eigenvalue**. In the above equation the eigenspace $$E_{\lambda} = \text{span}\{\vec{v}\}$$ because there is one eigenvector $$\vec{v}$$ corresponding to $$\lambda$$.

> **Proof:** Let $$\vec{v}$$ be an eigenvector of the matrix $$\textbf{A}$$. Then $$\textbf{A}\vec{v} = \lambda \vec{v}$$ for a unique $$\lambda$$. Now, let’s scale $$\vec{v}$$ by a constant $$c$$. $$\textbf{A}(c\vec{v}) = c(\textbf{A}\vec{v}) = c(\lambda \vec{v}) = \lambda (c\vec{v})$$ Therefore, $$\lambda$$ is still the associated eigenvalue of $$c\vec{v}$$.
