In Boyd & Vandenberghe's Convex Optimization, one can find two definitions for the ellipsoid.
$$ \mathcal{E} = \left \{ x \mid (x-x_c)^\mathsf{T}P^{-1}(x-x_c) \leq 1 \right\} $$
and
$$ \mathcal{E} = \left \{ x_c+Au \,|\, \| u \| \leq 1 \right \} $$
where $A=P^{1/2}$. I would like to know how to go from the first definition to the second and vice versa. In my doubts, I created a small program with a simple 2d ellipsoid with both definitions to verify that both definitions are the same. I obtained the following image.
If you are curios you can see the code in the following link.
This is how I convinced myself that both definitions of $\mathcal{E}$ are the same. But I'm not able to go from one definition to the other.
The only thing I could do is to pass from the second definition to
$$ \mathcal{E} = \left \{ x \mid \| A^{-1} (x-x_c) \|_2 \leq 1 \right \} $$
But I'm not sure if that helps of if it helps at all.
Given a vector $u \in \mathbb R^3$, the following statements are equivalent:
Indeed, with $x$ defined to be $x_c + A u$, we have $$ (x - x_c)^T P (x - x_c) = u^T A^T P A u = u^T A P A u = u^T P^{-1/2} PP^{-1/2} u = u^T u = \| u \|^2,$$ so certainly, $\| u \| \leq 1$ is true if and only if $(x - x_c)^T P (x - x_c) \leq 1$.
Therefore, $\left \{ x_c+Au \,|\, \| u \| \leq 1 \right \} $ is precisely the same set as $ \left \{ x \mid (x-x_c)^\mathsf{T}P^{-1}(x-x_c) \leq 1 \right\} $.