Question: What exactly is the matrix of a quadratic form? I have seen this notation occuring in a few papers (e.g. Siegel's unreadable German papers), with particular reference to the trace of a quadratic form. I'm at a loss as to what this means, and as a bonus question in passing I'd be interested if the trace of a quadratic form was interesting for an "obvious" reason (echoes of character theory, maybe?) 

For example what is the matrix of the quadratic form $x^2+y^2+z^2$? Or $x^2+xy+y^2$?

Answer: 
A quadratic form $q \colon V \to K$ on a vector space $V$ arises as "diagonal" of a symmetric bilinear form $\beta\colon V \times V \to K$ via $q(v) = \beta(v,v)$. Given $V$ has a basis $(v\_i)$, we can associate a matrix $B = (b\_{ij})$ with $\beta$, where $b\_{ij} = \beta(v\_i, v\_j)$. For $v = \sum \lambda\_i v\_i$, $w = \sum\mu\_i v\_i \in V$ we have then 
\[ \beta(v,w) = \sum\_{i,j} \lambda\_i b\_{ij} \mu\_j = (\lambda\_i)\_i^T B (\mu\_j)\_j, \]
so for $q$ we have
\[ q(v,v) = (\lambda\_i)^T B (\lambda\_i) \]
For your examples: 
\[
\begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\end{pmatrix}\text{ and } 
\begin{pmatrix} 1 & \frac 12 \\ \frac 12 & 1 \end{pmatrix}.
\]
For many authors, the matrix is actually double the ones given by martini and Hurkyl, so that the matrix is, quite simply, the Hessian matrix of the function, meaning the second partials. This is the more common behavior when discussing "even" lattices, as in most of [NEBE](http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/). The importance of doubling everything is to arrange that all inner products of vectors in the lattice be integers. The most important example here is the root lattices of Lie algebras. So, with that in mind see [PAGE $E\_8$](http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/E8.html) where "the" matrix is given as 

`4 -2 0 0 0 0 0 1
-2 2 -1 0 0 0 0 0
 0 -1 2 -1 0 0 0 0
 0 0 -1 2 -1 0 0 0
 0 0 0 -1 2 -1 0 0
 0 0 0 0 -1 2 -1 0
 0 0 0 0 0 -1 2 0
 1 0 0 0 0 0 0 2`

under the word GRAM.