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An ellipsoid in $\mathbf{R}^n$ has the following form

\begin{equation}\label{ellipsoid} \mathcal E = \left\{ x \mid (x - x_c)^T P^{-1} (x - x_c) \leq 1 \right\} = \left\{ x_c + Ax \mid \left \|x \right \|_2 \leq 1 \right\} \end{equation}

where $P = P^T \succ 0$ and matrix $A$ is square and nonsingular ($A=P^{1/2}$). The ellipsoid can also be defined as follows:

\begin{equation}\label{ellipsoid1} \mathcal E = \left\{ x \mid (x - x_c)^T P^{-1} (x - x_c) \right\} = \left\{ x \mid (x-x_c)^T E (x-x_c) \leq 1 \right\} \end{equation}

Here it suggests to formulate the ellipsoid equation as the following form:

\begin{equation} \mathcal E = \left\{ x \in \mathbb{R}^n \mid \left \| Ax-b \right \|_2 \leq 1 \right\} \end{equation} where $A = E^{1/2}$ and $b = E^{1/2}x_c$.

Problem: After expanding I was able to prove $A = E^{1/2}$. However, I feel like $b = E^{1/2}x_c$ seems to be wrong, could someone help me to find out what went wrong?

$(x-x_c)^tE(x-x_c) = x^tEx - x^tEx_c - x_c^tEx + c^tEc$

$(Ax-b)^t(Ax-b) = x^tA^tAx -x^tA^tb-b^tAx+b^tb$

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  • $\begingroup$ Linking to the PDF is bad manners. Please link to the DOI. $\endgroup$ Apr 2, 2022 at 8:16
  • $\begingroup$ Why do you say that? If you equate term by term the two expressions you wrote, this is precisely what you get. $\endgroup$
    – lcv
    Apr 2, 2022 at 8:26
  • $\begingroup$ $\| Ax-b\|_2$ is l2 norm, could put it as answer, I could not prove it myself. $\endgroup$
    – GPrathap
    Apr 2, 2022 at 8:30

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