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 $x^2+xy+y^2$?