Does the set of points $\{\mathbf{x}|\mathbf{x}=(\mathbf{v}^H\mathbf{R}\mathbf{v})\mathbf{v}\}$ represent a multidimensional ellipse?

Question

Given a unit-length column vector $\mathbf{v}$, does the set of points given by the length-scaled vector

$\{\mathbf{x}|\mathbf{x} = (\mathbf{v}^H\mathbf{R}\mathbf{v})\mathbf{v}\}$

represent a multidimensional ellipse? I guess it does since the projection of $\mathbf{x}$ onto the eigenvector $\mathbf{v}_k$ of $\mathbf{R}$ is its eigenvalue $\lambda_k$.

But how to prove it?