Noteworthy Uses:: [[product of linear map and adjoint is a positive semidefinite operator]] Proved By:: *[[Proved By|Crucial Dependencies]]* Intuition:: [[complex numbers|product of z and z conjugate is real]] ----- > [!proposition] Proposition. ([[product of linear map and adjoint is self-adjoint]]) > 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 [[self-adjoint]]. > [!proof]- Proof. ([[product of linear map and adjoint is self-adjoint]]) > Let $v,w \in V$. We have by definition of [[adjoint]] that > $\langle T^{\dagger}Tv,\,w \rangle = \langle Tv,Tw \rangle =\langle v,T^{\dagger}Tw \rangle$ and $\langle TT^{\dagger}v,w \rangle = \langle T^{\dagger}v, T^{\dagger}w \rangle = \langle v,TT^{\dagger}w \rangle ,$ which is what we wanted to show. ----- #### ---- #### 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 > ```