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 $x^2+xy+y^2$?
Best Answer
The matrix of a quadratic form $Q$ is the symmetric matrix $A$ such that
$$Q(\vec{x}) = \vec{x}^T A \vec{x}$$
For example,
$$x^2 + xy + y^2 = \left(\begin{matrix}x & y \end{matrix}\right) \left(\begin{matrix}1 & \frac{1}{2} \\ \frac{1}{2} & 1 \end{matrix}\right) \left(\begin{matrix}x \\ y \end{matrix}\right) $$