Unitary matrices and ellipsoid

Question

Let $\mathcal{S}$ be an ellipsoid in $\mathbb{R}^n$ and let $U\in\mathbb{R}^{n\times n}$ be a unitary matrix. Is it true that $U\mathcal{S}=\{y| \quad \exists x \in\mathcal{S} \quad s.t. \quad y=Ux\}$ is also an ellipsoid?

Answer 1

Yes, it's true. Since the ellipsoid can be represented by $xAx + bx + c = 0$ for a Matrix $A$, a vector $b$ and a scalar $c$, one can make the Transformation $x \rightarrow Ux$ and then it follows that the Matrix $A$ changes to $U^TAU$. Also $b \rightarrow bU$. The Eigenvalues of the Matrix $A$ (determining whether it is an ellipsoid or another form) are preserved under unitary transformations; hence all Points $y$ covering also an ellipsoid.