Let's consider the set of vectors $\textbf{x}$ such that:
$(\textbf{z}^T\textbf{x})^2\le \textbf{z}^TD\textbf{z}$
Here $D$ can be considered to be diagonal, and in any case is a matrix related to $\textbf{x}$. Also we can consider the set of all vectors $\textbf{z}$ normalized. I have to show that this set of vectors are points inside an n-dimensional ellipsoid.
Here my thoughts:
$(\textbf{z}^T\textbf{x})(\textbf{z}^T\textbf{x})\le \textbf{z}^TD\textbf{z}$
$\textbf{z}^T\textbf{x}\textbf{x}^T\textbf{z}\le \textbf{z}^TD\textbf{z}$
$\textbf{z}^T(\textbf{x}\textbf{x}^T-D)\textbf{z}\le 0$
Since $\textbf{x}\textbf{x}^T$ is a positive definite matrix, for the inequality $\textbf{z}^T\textbf{x}\textbf{x}^T\textbf{z}\le \textbf{z}^TD\textbf{z}$ the matrix $(\textbf{x}\textbf{x}^T-D)$ should be positive definite.
It follows that the matrix $(\textbf{x}\textbf{x}^T-D)$ should represent an ellipsoid and the set of points represented by $\textbf{z}$ should be inside the ellipsoid since the inequality is represented by $\le$.
Is it correct? I don't know if my considerations about the positive definiteness of the matrices are correct