---- Let $V$ be a [[vector space]]. > [!definition] Definition. ([[nilpotent linear operator]]) > A [[linear operator]] $\varphi \in \text{End}(V)$ is said to be **nilpotent** if $\varphi^{n}=0$ for some $n \geq 0$. ^definition > [!equivalence] > (Over $\mathbb{C}$) $\varphi$ is nilpotent if and only if all of its [[eigenvalue|eigenvalues]] are zero. [[Schur's Theorem]] gives a basis with respect to which $\varphi$ is [[strictly triangular matrix|strictly upper triangular]], $\varphi \in \mathfrak{n}_{n}$. ^equivalence > [!proof] Proof of Equivalence. > $0= \lambda(\varphi^{n})=\lambda^{n}(\varphi)$. For $\lambda \in \mathbb{C}$, $\lambda^{n}=0 \iff \lambda=0$. Conversely, if all eigenvalues of $\varphi$ are zero, then it has a [[basis]] with respect to which its [[matrix]] is strictly upper triangular. Strictly upper triangular matrices are nilpotent. > ^proof ---- #### ---- #### 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 > ```