Linear Map of an ellipsoid in $\mathbb{R}^N$ into another ellipsoid in $\mathbb{R}^n$, with $n<N$

Question

Starting from the closed set describing an ellipsoid in $\mathbb{R}^N$: $$\Omega_x = \{ x \in \mathbb{R}^N : (x-x_0)^T\Sigma_x^{-1}(x-x_0) \leq \varepsilon^2 \}$$ where $\Sigma_x \in \mathbb{R}^{N\times N}$ is a symmetric positive definite square matrix and $x_0 \in \mathbb{R}^N$, I need to find a way a to proove that its image through the linear map $\varphi: \mathbb{R}^N \rightarrow \mathbb{R}^n,\; n<N$, defined as the product with a full rank rectangular matrix: $$y = \varphi(x) := Px, \; P\in\mathbb{R}^{n\times N},\;\rho(P)=n$$ is still an ellipsoid, in $\mathbb{R}^n$ of course, defined in this way: $$\Omega_y = \varphi(\Omega_x) = \{y \in \mathbb{R}^n: y = Px, \;x \in \Omega_x\} = \{y \in \mathbb{R}^n : (y-y_0)^T\Sigma_y^{-1}(y-y_0) \leq \varepsilon^2 \}$$ with: $$y_0 = Px_0 \in \mathbb{R}^n$$ $$\Sigma_y = P\Sigma_yP^T \in \mathbb{R}^{n\times n}$$