[[Rational lattice]]
# Self-dual rational lattice

A [[rational lattice]] is **self-dual** iff it is its own [[Dual of a rational lattice|dual lattice]], or equivalently it is [[Rational lattice#^integral]] and [[Unimodular lattice|unimodular]]. #m/def/geo

> [!check]- Proof
> Note for a unimodular integral matrix $G$ we have $G\mathbb{Z}^n = \mathbb{Z}^n$.
> 
> Let $A$ be a basis matrix for $L$ so that $\tp A A = G$,
> so the basis matrix $A^\circ$ of $L^\circ$ is $A G^{-1}$.
> Now assuming $G$ is unimodular, so is $G^{-1}$ and we have
> $$
> \begin{align*}
> L^\circ = A^\circ \mathbb{Z}^n = AG^{-1} \mathbb{Z}^n = A\mathbb{Z}^n = L
> \end{align*}
> $$
> as required. <span class="QED"/>

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