Gram Matrix

Given $m$ points with $n$-D vector coordinates ${\bf v}_i$, let ${\hbox{\sf M}}$ be the $n\times m$ matrix whose $j$th column consists of the coordinates of the vector ${\bf v}_j$, with $j=1$, ..., $m$. Then define the $m\times m$ Gram matrix of dot products $a_{ij}={\bf v}_i\cdot{\bf v}_j$ as

$${\hbox{\sf A}}={\hbox{\sf M}}^{\rm T}{\hbox{\sf M}},$$

where ${\hbox{\sf A}}^{\rm T}$ denotes the Transpose. The Gram matrix determines the vectors ${\bf v}_i$ up to Isometry.