Category:Definitions/Determinants of Point Lattices

This category contains definitions related to Determinants of Point Lattices.
Related results can be found in Category:Determinants of Point Lattices.

Let $\R^m$ be the $m$-dimensional real Euclidean space.

Let $\BB = \set {\mathbf v_1, \mathbf v_2, \ldots, \mathbf v_n}$ be a linearly independent set of vectors of $\R^m$.

Let $\map \LL \BB$ be the point lattice in $\R^m$ whose basis is $\BB$.

Let $\MM = \begin {pmatrix} \mathbf v_1 & \mathbf v_2 & \cdots & \mathbf v_n \end {pmatrix}$ be the square matrix formed by writing the coordinates of $\mathbf v_1, \mathbf v_2, \ldots, \mathbf v_n$ in rows.

The determinant $\DD$ of $\map \LL \BB$ is the absolute value of the determinant of $\MM$:

$\DD = \size {\det \begin {pmatrix} \mathbf v_1 & \mathbf v_2 & \cdots & \mathbf v_n \end {pmatrix} }$