let's say we have a system with vector $q_{(t)}$ representing the degrees of freedom (DoF), and state vector $ x_{(t)} = \left \{ \begin{array}{c c} q_{(t)} \\ \dot{q_{(t)}} \end{array} \right \}$ which includes each DoF and it's corresponding velocity. The system dynamics are often represented implicitly as follows (I'll omit the dependence on time of the state variables from now on) :
$$ M_{(q)} \dot{x} = H_{(q,\dot{q})} = H_{(x)} $$
where $M_{(q)}$ is the 'Mass-matrix'. In the case $M = I$, the identity matrix, the system is explicit. The system can be rendered explicit quite simply as follows:
$$ \dot{x} = M_{(q)}^{-1}H_{(x)} = \hat{H}_{(x)} $$ To linearise around an equilibrium point $x_0$, we would take the Jacobian of $H_{(x)}$, and the system dynamics would be, in the explicit case, approximated as $$ \dot{x} = J_{(x_0)} (x-x_0) \\ J_{(x_0)} = \nabla_xH_{(x)}|_{(x=x_0)} $$
My question is, how to deal with linearisation in the implicit case? Are the following cases equivalent?
$$ \dot{x} = M_{(q)}^{-1}J_{(x)} = M_{(q)}^{-1}\nabla_xH_{(x)} \stackrel{?}{=} \nabla_x\hat{H}_{(x)} =\nabla_x[M_{(q)}^{-1}H_{(x)}] $$
In other words, is it imperative to render the EoM explicit first? In even simple systems where a viewer can quite easily imagine the dynamics when looking at the implicit form, the explicit form becomes extremely long, unintuitive, not to mention very difficult to compute. Are there any workarounds that avoid solving $[M_{(q)}^{-1}H_{(x)}]$?