I'm trying to develop a solution for a minimization problem for a project. This requires me to use the following equation relating $\dot X$ and $\dot Q$ through the Jacobian matrix. If needed, Q is a set of three limb angles while X encapsulates the corresponding endpoint Cartesian coordinates.
$$\dot X = J \dot Q $$ Where $$ J = \left[ \begin{matrix} \frac{dq_1}{dx_1} & \frac{dq_2}{dx_1} & \frac{dq_3}{dx_1} \\ \frac{dq_1}{dx_2} & \frac{dq_2}{dx_2} & \frac{dq_3}{dx_2} \\ \end{matrix} \right] $$
Now, In my solution, I end up with the following expression $$ J^TJ\dot Q = J^T\dot X $$ Where $J^TJ$ is a symmetric square matrix. Is it possible now to conclude through some kind of positive definiteness properties of this matrix that it is invertible? If so, how? This one step should enable me to obtain an explicit relation between $\dot X$ and $\dot Q$.