Dear Convex Analysis Experts,
I am bewildered by several definitions all around me. The question in particular here is regarding the relationship between "Ellipsoid", "Quadratic Form" (and "Numerical Range"). Could you please help me to understand whether they are same thing but just different names or there are some notably technical differences?
Please correct my understanding of these definitions, if I am wrong. Apologies in advance, if I have mixed up. I hope it is OK to use the set definition rather than function per se.
"(General) Ellipsoid": $\mathcal{C}_E = \left\{\mathbf{x} \in \mathbb{C}^n : \ \left(\mathbf{x} - \mathbf{x}_0\right)^{\rm H} \mathbf{A} \left(\mathbf{x} - \mathbf{x}_0\right) \leq 1 \right\}$? The conjugate Hermitian transpose is denoted as $(\cdot)^{\rm H}$. The matrix $\mathbf{A} \in \mathbb{C}^{n \times n}$. Does it need to be positive semidefinite? For instance, in 2D space, it is like an ellipse. Correct?
"(General) Quadratic Form": $\mathcal{C}_Q = \left\{\mathbf{x} \in \mathbb{C}^n : \mathbf{x}^{\rm H} \mathbf{A} \mathbf{x} -2 \mathfrak{R}\left\{\mathbf{b}^{\rm H} \mathbf{x}\right\} \leq c \right\} \equiv \left\{\mathbf{x} \in \mathbb{C}^n : \left(\mathbf{x} - \mathbf{x}_0\right)^{\rm H} \mathbf{A} \left(\mathbf{x} - \mathbf{x}_0\right) \leq c^\prime \right\}$. If $c^\prime=1$, then quadratic form boils down to ellipsoid. Correct?
So, I see a thin difference between "Ellipsoid" and "Quadratic form". And, I guess, one could say that the "Quadratic form" is more generalized form than "Ellipsoid"?
Thank you so much in advance