$$ \newcommand{\qed}{\tag*{$\square$}} \newcommand{\span}{\operatorname{span}} \newcommand{\dim}{\operatorname{dim}} \newcommand{\rank}{\operatorname{rank}} \newcommand{\norm}[1]{\|#1\|} \newcommand{\grad}{\nabla} \newcommand{\prox}[1]{\operatorname{prox}_{#1}} \newcommand{\inner}[2]{\langle{#1}, {#2}\rangle} \newcommand{\mat}[1]{\mathcal{M}[#1]} \newcommand{\null}[1]{\operatorname{null} \left(#1\right)} \newcommand{\range}[1]{\operatorname{range} \left(#1\right)} \newcommand{\rowvec}[1]{\begin{bmatrix} #1 \end{bmatrix}^T} \newcommand{\Reals}{\mathbf{R}} \newcommand{\RR}{\mathbf{R}} \newcommand{\Complex}{\mathbf{C}} \newcommand{\Field}{\mathbf{F}} \newcommand{\Pb}{\operatorname{Pr}} \newcommand{\E}[1]{\operatorname{E}[#1]} \newcommand{\Var}[1]{\operatorname{Var}[#1]} \newcommand{\argmin}[2]{\underset{#1}{\operatorname{argmin}} {#2}} \newcommand{\optmin}[3]{ \begin{align*} & \underset{#1}{\text{minimize}} & & #2 \\ & \text{subject to} & & #3 \end{align*} } \newcommand{\optmax}[3]{ \begin{align*} & \underset{#1}{\text{maximize}} & & #2 \\ & \text{subject to} & & #3 \end{align*} } \newcommand{\optfind}[2]{ \begin{align*} & {\text{find}} & & #1 \\ & \text{subject to} & & #2 \end{align*} } $$
An ellipsoid centered at is a set
where is a self-adjoint, positive definite matrix. The lengths of the semi-axies of are given by , where the are the eigenvalues of To see why is an ellipsoid when , write , where the columns of are orthonormal eigenvectors of , and notice that
where . In particular, if , then
An alternate representation of an ellipsoid is
If is symmetric and positive definite, then taking gives the ellipsoid defined in the first representation. To see this for the case when , note that taking gives , which in turn equals .