---- $\mathbb{F}$ denotes $\mathbb{R}$ or $\mathbb{C}$. Let $V$ be a [[vector space]] over $\mathbb{F}$. > [!definition] Definition. ([[minimal polynomial]]) > The **minimal polynomial** of $T \in$[[vector space of operators]] is the *unique* [[monic polynomial]] $\mu$ of smallest degree such that $\mu(T)=0$. > [!justification] > The [[Cayley-Hamilton Theorem]] tells us that the [[characteristic polynomial]] $\chi$ of $T$ satisfies $\chi(T)=0$. It makes sense to wonder if there are other [[polynomial]]s with the property. In some sense, the **minimal polynomial** of $T$ is the 'simplest' such. > \ > It also makes sense to wonder when $\chi=\mu$. This happens, e.g., with [[companion matrix|companion matrices]]. > \ > ---- #### ---- #### 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 > ```