Recently I've been studying quadratic forms $p^TBp$ algebraically, but I would like to be able to visualize them (the ones from $\mathbb{R}^2 \to \mathbb{R}$ of course). I have done some experiments for myself using the online grapher math3d, but I don't know if I captured all the possibilities.
- Positive definite quadratic forms seem to be upward-opening parabolas rotated around the line $\{(0, 0, z)|z \in \mathbb{R} \}$, which are then streched or shrinked along the $x$ and $y$ axes (not sure if along other directions as well)
Ex:
$$\textbf{x}^T
\begin{bmatrix}
1 & 1\\
1 & 4
\end{bmatrix}\textbf{x} =
x^2+2xy+4y^2$$
- Positive semidefinite quadratic forms seem to be rotated and streched versions of the graph of $f(x, y) = x^2.$
Ex: $$\textbf{x}^T \begin{bmatrix} 1 & 2\\ 2 & 4 \end{bmatrix} \textbf{x} = x^2+4xy+4y^2$$
Negative definite and semifinite forms are just negatives of positive definite and semidefinite forms.
Indefinite quadratic forms seem to be all saddles.
Ex:$$ \textbf{x}^T \begin{bmatrix} 1 & 3\\ 3 & 4 \end{bmatrix} \textbf{x}^T= x^2+6xy+4y^2 $$
- I know it's only worth looking at quadratic form with symmetric matrices, since for any matrix $A$ there exists a symmetric matrix $B$ such that $\forall v: v^TAv = v^TBv$
My question is, did I capture all the possibilities? Is there anything else helpful, or other "common knowledge", that I should know about the graphs of quadratic forms?
Thank you very much!