SUMMARY
Consider the $\mathbb{dz}$ function, modeled as a switching system \begin{equation} \mathbb{dz}(w(t)) = \left\{\begin{matrix} -\rho, & w(t) > \rho \\ 0, & \left | w(t) \right | \leq \rho \\ \rho, & w(t) < -\rho \\ \end{matrix}\right. \end{equation}
When $|w(t)| < \rho$, the function is inside a symmetric set defined by \begin{equation} \mathbb{S} = \left \{ w(t) \in \mathbb{R}^{n}; w(t)^{\top}w(t) \leq \rho^{2} \right \} \end{equation}
PROBLEM
My problem is the argument of dead-zone, which is in norm form as follows \begin{equation} w(t) = \left \| x(t)\right \|_{G}^{2} \Rightarrow x(t)^{\top}Gx(t) \end{equation}
I need to convert the norm in a product form like \begin{equation} x(t)^{\top}Gx(t) \xrightarrow[magic]{} Sx(t) \end{equation}
TRIED SOLUTION
I've tried to apply Cholesky decomposition \begin{equation} x(t)^{\top}Gx(t) \Rightarrow x(t)^{\top}G_1^{\top}G_1x(t) \Rightarrow |G_1x(t)| \end{equation}
From that point, I didn't find a mathematical way to get $Sx(t)$. Is there a way to work on that problem ?