Gram matrix (reviewed)

Fixing a basis $\{v_1,...,v_n\}$ for a space $V$, the Gram matrix is the $n\times n$ symmetric matrix given by $[B(v_i,v_j)]$. Similarly, if we fix a basis $\{w_1,...,w_n\}$ for a lattice $L$, then it is also a basis for $V$, and the Gram matrix for $L$ is given by $[B(w_i,w_j)]$.

For example, given the integral lattice $\mathbb Z^3$ on the Euclidean space $\mathbb{R}^3$, in the standard basis $\{e_1,e_2,e_3\}$ the lattice (and hence the space) has a Gram matrix \[ \begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}. \]

Although Gram matrices are only unique up to a choice of basis, each lattice in the database is identified by a single Gram matrix. By entering a Gram matrix, $G$, into the search field, the database will return an isometric lattice whose Gram matrix, $A$, is equivalent to $G$ via conjugation by some transition matrix, $T$ (that is, $A=T^tGT$), provided the lattice is in the database.