> [!NOTE] Definition (Linear Operator) > Let $V$ be a [[Vector Space|vector space]]. A linear operator on $V$ is a [[Linear Map|linear map]] $V\to V.$ # Properties > [!NOTE] Definition (Eigenpairs) > Consider a linear operator $\varphi:V \to V.$ We say that $\lambda\in\mathbb{R}$ is an [[Eigenpair|eigenvalue]] of $\varphi$ if there exists some nonzero $v\in V$ such that $\varphi(v)=\lambda v$We call $v$ an **eigenvector** of $\varphi$ corresponding to eigenvalue $\lambda.$ > >Let $A\in \mathbb{R}^{n\times n}$ be a [[Matrix|matrix]]. We say that $\lambda\in \mathbb{R}$ is an **eigenvalue** of $A$ if there exists some nonzero $\underline{v} \in \mathbb{R}^{n}$ such that $A\underline{v}=\lambda \underline{v}$We call $\underline{v}$ an **eigenvector** of $A$ corresponding to the eigenvalue $\lambda.$