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: $$ bx^2 + ay^2 \leq ab\\ \Rightarrow \frac{ax^2}{b} + \frac{by^2}{a} \leq 1 $$
$A$ 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).