Let $S^n$ define a symmetric matrix, $S=\{B: B=B^T\}$. We associate with each $A\in S^n_{++}$ an ellipsoid centered at the origin given by:
\begin{align}
\epsilon_A = \{x : x^T A^{-1}x\leq 1\}.
\end{align}
I am kinda a novice at linear algebra, so I am confused and don't really understand why the point $x$ is being multiplied twice by the inverse of $A$. Could anybody please elaborate on the notation here and how this actually creates an ellipsoid?
EDIT:
Looking at the $n=2$ case,
$$
A=
\begin{bmatrix}
a && c\\
c && b
\end{bmatrix} \Rightarrow
A^{-1}=
\begin{bmatrix}
b/(ab-c^2) && -c/(ab-c^2)\\
-c/(ab-c^2) && a/(ab-c^2)
\end{bmatrix}\\
x =
\begin{bmatrix}
x\\
y
\end{bmatrix}\\
\begin{bmatrix}
x,y
\end{bmatrix}
\begin{bmatrix}
b/(ab-c^2) && -c/(ab-c^2)\\
-c/(ab-c^2) && a/(ab-c^2)
\end{bmatrix}
\begin{bmatrix}
x\\
y
\end{bmatrix}\\
=
\begin{bmatrix}
x,y
\end{bmatrix}
\begin{bmatrix}
xb/(ab-c^2)-yc/(ab-c^2)\\
ya/(ab-c^2)-xc/(ab-c^2)
\end{bmatrix}\\
= \frac{bx^2+ay^2-c(x+y)}{ab-c^2}\leq1\\
\Rightarrow bx^2 + ay^2 \leq (ab-c^2)+c(x+y)\\
\Rightarrow bx^2 + ay^2 – c(x+y)\leq -c^2+ab
$$
Setting $c=0$ we get the equation for an ellipse:
$$
bx^2 + ay^2 \leq ab\\
\Rightarrow \frac{ax^2}{b} + \frac{by^2}{a} \leq 1
$$
Best Answer
$A$ can be written as $A=U \Lambda U^T$ where $U$ is orthogonal and $\Lambda$ diagonal.
Note that $\epsilon_\Lambda = \{ y | \sum_k {1 \over \lambda_k} y_k^2 \le 1 \}$ and so $\epsilon_A = U \epsilon_\Lambda$ (that is, a rotation of the $\Lambda$ ellipsoid).