The [[Gram Matrix]] of some [[Matrix]] $A$ is the [[Matrix Product|product]] of $A$ with its [[Matrix Transpose|Transpose]]. $\huge G_{A} = A A^{\intercal} $ This can also be expressed as the [[Outer Product]] between $A$ and itself. $ \huge G_{A} = A\otimes A $ The [[Gram Matrix]] of any [[Matrix]] $A$ is *always* a [[Symmetric Matrix]]. If $A$ is [[Injective]], then the [[Gram Matrix]] is [[Positive Definite Matrix|Positive Definite]], however it will *always* be [[Positive Semidefinite Matrix|Positive Semidefinite]]. >[!example] Proof >Suppose $A^{\intercal}A \vec v = \lambda \vec v$, $\lambda <0$, >$ \huge \begin{align} >\vec v ^{\intercal}A ^{\intercal} A\vec v &= \vec v^{\intercal}(\lambda \vec v) \\ >&=\lambda \vec v^{\intercal} \vec v \\ >&= \lambda \norm{ \vec v } ^{2} < 0 >\end{align} $ >Which is a contradiction, therefore $A$ is [[Positive Semidefinite Matrix|Positive Semidefinite]].