Let $L$ be a $k$-dimensional lattice in $\mathbb{R}^n$. The *covolume*
$\hbox{CoVol}(L)$ of $L$ is the $k$-dimensional volume of a
fundamental domain for $L$, i.e., the volume of the parallelopiped
spanned by a $\mathbb{Z}$-basis for $L$. Now let $m<n$, let $A$ be an
$m$-by-$n$ matrix with coordinates in $\mathbb{Z}$ and with
$\hbox{rank}(A)=m$, and let $\hbox{Minor}(A)$ denote the set of
$m$-by-$m$ minors of $A$. I'd like a reference for the following two
formulas, which must be standard results, but I didn't find them on a
quick look at a couple of references.
$$
 (1)\qquad
 \hbox{CoVol}\bigl(\hbox{Row Span}(A)\bigr) = \sqrt{\det(A\cdot {}^t\!A) }
 = \sqrt{\sum\_{B\in\text{Minor}(A)} \det(B)^2}.
$$
Let $\phi\_A:\mathbb{Z}^n\to\mathbb{Z}^m$ be the linear transformation
$\phi\_A(\boldsymbol{w})=\boldsymbol{w}A$ associated to the matrix $A$.
Then $\hbox{Ker}(\phi\_A)$ is an $(n-m)$-dimensional lattice whose covolume
is given by the formula
$$
 (2)\qquad
 \hbox{CoVol}\bigl(\hbox{Ker}(\phi\_A)\bigr)
 = \frac{\hbox{CoVol}\bigl(\hbox{Row Span}(A)\bigr)}
 {\gcd\bigl\{\det(B) : B\in\hbox{Minor}(A)\bigr\}}.
$$



Covolume of the row span of a matrix and of the kernel of a matrix
Equality (2) can be proved through the [Smith Normal Form](http://en.wikipedia.org/wiki/Smith_normal_form) of $A$. I am pretty sure there is a simpler proof then the one below.


One can decompose $A$ into: $A = U D V^t$ where $D = ( \hat{D}\,\, \left|\right.\,\, {0}\_{m \times (n-m)})$, $\hat{D}$ is diagonal $m \times m$, $\hat{d}\_{ii} = \delta(i)/\delta(i-1)$, and $\delta\_i$ is the gcd of the $i \times i$ minor determinants of $A$. The matrices $U$, $V$ are unimodular (of dimensions $m \times m$ and $n \times n$, respectively). 


Since $m < n$, partitioning $V= \left(\begin{array}{c} V\_1 \\ V\_2 \end{array}\right)$, where $V\_1$ has dimensions $m \times n$, gives us $A = U \hat{D} V\_1$, from where we have


$$\mbox{CoVol}(\mbox{RowSpan}(A))= \sqrt{\det{ A A^t}} = \det(\hat{D})\sqrt{\det(V\_1 V\_1^t)}.$$
But ${\det(\hat{D})} = \delta\_m = \gcd(\det B: B \in\mbox{Minor}(A))$. Hence it is only left to prove that $\sqrt{\det(V\_1 V\_1^t)} = \mbox{CoVol}(\ker(\phi\_A))$ (I am (re)-defining $\ker(\phi\_A) = \left\{w \in\mathbb{Z}^n: Aw = 0\right\}$). It is not hard to see that $\ker(\phi\_A) = \mathbb{Z}^n \cap V\_1^\perp$, where $V\_1^\perp$ is the subspace orthogonal to the rows of $V\_1$. From this characeterisation (p.166 of Conway and Sloane's book [1](http://en.wikipedia.org/wiki/Smith_normal_form)), we show that $\sqrt{\det(V\_1 V\_1^t)} = \mbox{CoVol}(\ker(\phi\_A))$.


[1](http://en.wikipedia.org/wiki/Smith_normal_form) Sphere Packings, Lattices and Groups, 3rd Ed.


