----- > [!proposition] Proposition. ([[Gram matrices are positive semidefinite]]) > [[Gram matrix|Every Gram matrix]] is a [[positive definite matrix|positive semidefinite matrix]]. > [!proof]- Proof. ([[Gram matrices are positive semidefinite]]) > Let [[inner product space]] an [[inner product space|inner product space]], and $W$ an [[inner product space|inner product space]]. Denote by $\mathcal{B}_{V}$ resp. $\mathcal{B}_{W}$ the standard [[basis]] of $V$ resp. $W$. Let the [[inner product]]s be the [[Euclidean inner product]]. \ Recall that any [[matrix]] $A$ is [[positive definite matrix|positive semidefinite]] if and only if $\langle Tv, v\rangle > 0 \ \fa v \in V$, where $T \in$ [[vector space of operators on a vector space]] is the linear [[linear operator]] given by $Tv=Av$ $(*)$: $\begin{align}T \text{ is PSD} \iff & \langle Tv,v \rangle \geq0 \\ \iff & \langle Av,v \rangle \geq 0 \\ \iff &\langle v,Av \rangle \geq 0 \\ \iff& v^{\top}Av\geq 0 \\ \iff & A \text{ is PSD}.\end{align}$ So, let $T \in$ [[vector space of operators on a vector space]] such that $T(v)=Gv$; we will show that $T$ is a [[positive semidefinite operator|positive semidefinite operator]]. \ Since [[every Gram matrix is the matrix of R*R w.r.t. standard basis for some R]], fix $R \in$ [[vector space of linear maps between two vector spaces]] such that $T=R^{\dagger}R$. The result now follows immediately from [[product of linear map and adjoint is a positive semidefinite operator]] and $(*)$. ----- #### ---- #### 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 > ```