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.

**Authors:**

**Knowl status:**

- Review status: reviewed
- Last edited by Kiran S. Kedlaya on 2018-06-27 09:25:30

**Referred to by:**

- dq.lattice.source
- lattice.definition
- lattice.determinant
- lattice.hermite_number
- lattice.isometry
- lattice.postive_definite
- lattice.primitive
- lattice.theta
- lmfdb/lattice/templates/lattice-index.html (line 88)
- lmfdb/lattice/templates/lattice-search.html (line 13)
- lmfdb/lattice/templates/lattice-single.html (line 58)

**History:**(expand/hide all)

- 2018-06-27 09:25:30 by Kiran S. Kedlaya (Reviewed)