How to derive the following Matrix form of the ellipse equation?

Question

In convex optimization book (By Stephen Boyd) it is mentioned that an set of points which is bounded by an elliptical shape (and hence ellipsoid) can be written as follows $$\{x| (x-x_c)^{T}P^{-1}(x-x_c)\leq r\}$$ where $P^{-1}$ is some positive definite matrix. The general expression for the ellipse in 2 dimension is as follows $$\frac{(x-x_c)^2}{a^2}+\frac{(y-y_c)^2}{b^2}=h$$ I want to know how derive the first equation from the second equation. Any help in this regard will be much appreciated. Thanks in advance.

Answer 1

You are wrong if think that your last expression is the general form of an ellipsis. Howver, given such an ellpsis, the region bounded by it is$$\left\{(x,y)\in\mathbb{R}^2\,\middle|\,\frac{(x-x_c)^2}{a^2}+\frac{(y-y_c)^2}{b^2}\leqslant h\right\}$$and this set is equal to$$\left\{(x,y)\in\mathbb{R}^2\,\middle|\,\begin{pmatrix}x-x_c&y-y_c\end{pmatrix}\begin{pmatrix}\frac1a&0\\0&\frac1b\end{pmatrix}\begin{pmatrix}x-x_c\\y-y_c\end{pmatrix}\leqslant h\right\}.$$