Moving between different ellipse representations

Question

I have a representation of an ellipse that is the affine transform of the unit ball, $\|Ax + b\| <= 1$.

My question is, how can I change this ellipse representation? I would like to have it in the form $x^T A x + 2b^Tx +c <= 0$ where $A$ is symmetric. I believe that the $A$ and $b$ in equation 1 are different to the $A$ and $b$ in equation 2.

Thanks for your help.

Answer 1

It's rather obvious that if $\vert\vert Ax + b \vert\vert \leqslant 1$, then $\vert\vert Ax + b \vert\vert^2 \leqslant 1$. Here we go: $\vert\vert Ax + b \vert\vert^2 = (Ax+b, Ax+b) = (Ax+b)^{T}(Ax+b) = (x^{T}A^{T} + b^{T})(Ax+b) = x^{T}(A^{T}A)x+ x^{T}A^{T}b + b^{T}Ax + b^{T}b=x^{T}(A^{T}A)x + 2(b^{T}A)x + (b^{T}b) = x^{T} \tilde{A} x + 2\tilde{b}^{T}x + \tilde{C}$. This is how you can move from one representation to another.