---- > [!definition] Definition. ([[linear projector]]) > A [[linear map]] $P \in \hom(V,V)$ is called a **projector** if it is [[idempotent]]: $P^{2}=P$. > > > [!equivalence] > $P$ is a projector iff it is [[diagonalizable]] with [[eigenvalue]]s all $0$ or $1$. ^equivalence > [!warning] Warning. > Although the [[projection of a point onto a set|projection of a point in a vector space onto a linear subspace]] [[minimum distance to a subspace is the orthogonal projection onto it|is an orthogonal projection]], the [[matrix]] of a projector is not necessarily an [[orthogonal projection matrix]]. The counterexamples are exactly the non-[[conjugate symmetric|Hermitian]] [[idempotent]] [[matrix|matrices]] ([[oblique projection matrix|oblique projection matrices]]). > ![[CleanShot 2023-11-18 at [email protected]]] ---- #### ---- #### 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 > ```