My question is about the $k$-dimensional volume of the simplex spanned by the origin together with $k$ vectors stored in an $k \times d$-matrix A. I found two references saying that this volume is $\frac{1}{j!} \cdot \sqrt{\det A A^T}$, and one reference that said it was $\frac{1}{(j!)^2} \cdot \det A A^T$. The first version seems to be consistent with the hints on the wikipedia-article on determinants, but I was hoping for a book or publication to back up that this is correct. So my question is:
Is the first version correct, and could you point out a reference / explanation for this fact? Thanks in advance.
The first version is correct, it has the right scaling. The volume scales as $L^k$ where $L$ is the typical "length" of the object. Then $AA^\top$ scales as $L^2$ and $det(AA^\top)$ scales as $L^{2k}$, so you need the square root.
But I can't give the book reference.