----- > [!proposition] Proposition. ([[product of linear map and adjoint is a positive semidefinite operator]]) > Let $T \in$ [[vector space of linear maps between two vector spaces]], where $V,W$ are each [[inner product space|inner product spaces]]. Then $T^{\dagger}T$ and $TT^{\dagger}$ are each [[positive semidefinite operator]]s. > [!proof]- Proof. ([[product of linear map and adjoint is a positive semidefinite operator]]) > (Recall that [[product of linear map and adjoint is self-adjoint]]; so the requirement that $T^{\dagger}T$ and $TT^{\dagger}$ are [[self-adjoint]] is immediately satisfied.) > Using the definition of [[adjoint]] we have for arbitrary $v \in V$ that $\langle T^{\dagger}Tv , v\rangle = \langle Tv, Tv \rangle \geq 0 $ by **positive definiteness** of the [[inner product]]. Hence the definition of [[positive semidefinite operator]] has been satisfied for $T^{\dagger}T$. \ Similarly, $\langle TT^{\dagger}v, v \rangle = \langle T^{\dagger}v, T^{\dagger}v \rangle \geq 0 $ for all $v \in V$. (This is one of those notes that gets used so much it is not referenced) ----- #### ---- #### 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 > ```