Properties:: [[every Gram matrix is the matrix of R*R w.r.t. standard basis for some R]] [[Gram matrices are positive semidefinite]] Sufficiencies:: *[[Sufficiencies]]* Equivalences:: *[[Equivalences]]* ---- > [!definition] Definition. ([[Gram matrix]]) > The **Gram matrix** of a set of [[vector]]s $v_{1},\dots,v_{n}$ in an [[inner product space|inner product space]] [[inner product space]] over $\rr$ or $\cc$ is the [[conjugate symmetric]] matrix whose entries are given by the (Euclidean) [[inner product]] $G_{ij}=\langle v_{i}, v_{j} \rangle$. That is, $G=V^{\dagger}V$, where $V$ is the [[matrix]] whose columns are $v_{1},\dots,v_{n}$. > > > $G = \begin{bmatrix} > \langle X_{:,1}, X_{:,1} \rangle & \langle X_{:,1}, X_{:,2} \rangle & \cdots & \langle X_{:,1}, X_{:,n} \rangle \\ > \langle X_{:,2}, X_{:,1} \rangle & \langle X_{:,2}, X_{:,2} \rangle & \cdots & \langle X_{:,2}, X_{:,n} \rangle \\ > \vdots & \vdots & \ddots & \vdots \\ > \langle X_{:,n}, X_{:,1} \rangle & \langle X_{:,n}, X_{:,2} \rangle & \cdots & \langle X_{:,n}, X_{:,n} \rangle > \end{bmatrix}$ > > [!justification]- > That **Gram matrices** are [[conjugate symmetric]] is immediate since [[inner product]]s are. ---- #### ---- #### 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 > ```