We have a non-linear ODE of the form
$ \dot x= f(x, h(x))$
where $g(x, h(x))=0$, $h: X\subseteq\mathbb R^n\to \mathbb R^m$, $g:\mathbb R^n\times \mathbb R^m\to \mathbb R^m$, $f:\mathbb R^n\times \mathbb R^m\to \mathbb R^n$ are smooth, could anyone tell me how the matrix of linearized ODE is formed as: $A=D_xf-D_yf(D_yg)^{-1}D_xg)_{|(x^*, h(x^*))}$ where $D_x$ denote the Jacobian w.r.t $x$. and $x^*$ is a stable point. Thank you! If you could give me a proper reference about how to find linearization of such ODE that would be nice too.
You might want to read up on the Schur-complement. Or linearize the DAE system directly \begin{align} \dot u &=D_xf(x^*,y^*)u+D_yf(x^*,y^*)v,\\ 0&=D_xg(x^*,y^*)u+D_yg(x^*,y^*)v, \end{align} $y^*=h(x^*)$, and use the second equation to eliminate $v$ from the first equation.