----- > [!proposition] Proposition. ([[orthogonal projection matrix]]) > A matrix $P$ is called an **orthogonal projection matrix** if $P$ is both [[idempotent]] and [[conjugate symmetric|hermitian]]. \ Let [[inner product space]] be an [[inner product space|inner product space]]. Unsurprisingly, the [[matrix]] of an [[orthogonal projection]] $P_{U}$ onto a [[linear subspace]] $U$ of $V$ is an **orthogonal projection matrix**, and every [[orthogonal projection matrix]] represents an [[orthogonal projection]] $P_{U}$. If $A$ is a [[matrix]] such that $\im A=U$, then $P_{U}=P_{\im A}=A A^{+}$. If the columns of $A$ form a basis for $U$, then they are [[linearly independent]] and we can further say $P_{U}=A(A'A)^{-1}A'$. This is derived using the [[Moore-Penrose pseudoinverse]] [[least squares]] solution. \ A nontrivial [[matrix]] $P$ is an **orthogonal projection matrix** if and only if it can be written as $UU'$ for some [[semi-isometric matrix|semi-unitary matrix]] $U$. \ ![[CleanShot 2023-10-07 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 > ```