---- > [!definition] Definition. ([[eigenvalue]]) > An [[eigenvector]] of a [[linear map]] $T \in$ [[vector space of operators on a vector space]] is an *nonzero* [[vector]] $v \in V$ such that $T(v) = \lambda v$ for some [[scalar]] $\lambda \in \mathbb{F}$. $\lambda$ is called the **eigenvalue** of the [[eigenvector]] $v$. > [!basicexample] >Take the [[derivative|differentiation map]] $D$ of $\mathcal{C^\infty}$, the [[vector space]] of single-variable [[smooth]] functions. Since $D \ e^{\lambda x} = \lambda e^{\lambda x}$ for all $\lambda \in \mathbb{R}$, we see that *every [[real numbers|real number]]* is an **eigenvalue** of $D$, with corresponding [[eigenvector]]s $f_\lambda (x) = e^{\lambda x}$. > [!equivalence] > # Equivalent Formulations > [[eigenvalue equivalencies]] > [[eigenvalue iff solves characteristic polynomial]] ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```