Suppose I have a 2D dynamical system with
$$\frac{dx}{dt} = f(x, v), \hspace{5mm} \frac{dv}{dt} = g(x, v)$$
My Jacobian is then given by $\begin{pmatrix} f_x & f_v \\ g_x & g_v\end{pmatrix}$.
I could then calculate the eigenvalues of this to get my stability.
However, what I could have labelled the equations the other way around and my stability matrix would become $\begin{pmatrix} g_x & g_v \\ f_x & f_v\end{pmatrix}$. In fact, there are many arrangements of this Jacobian I can have: $\begin{pmatrix} g_v & g_x \\ f_x & f_v\end{pmatrix}$, $\begin{pmatrix} g_x & f_x \\g_v & f_x\end{pmatrix}$ and so on…
So how do I know how to arrange my Jacobian?
For some of the above arrangements, we are talking about an interchanging of two rows of the matrix, and that does not leave the eigenvalues unchanged. Hence so stability of my system will change.
Of course, just calling them $f$ and $g$ is arbitrary and from what I can see at the moment, if I switch the labels, the dynamics change which should obvious not be the case.
Best Answer
The system is given by
$$\dot{x}=f(x,v)$$ $$\dot{v}=g(x,v).$$
The idea behind the Jacobian is that we use the Taylor expansion of the functions $f$ and $g$ evaluated at the point $x_0$ and $y_0$. The expansions are given by
$$f(x,v)\approx f(x_0,v_0)+\left.\dfrac{\partial f}{\partial x}\right|_{x=x_0,v=v_0}(x-x_0)+\left.\dfrac{\partial f}{\partial v}\right|_{x=x_0,v=v_0}(v-v_0)$$ $$g(x,v)\approx g(x_0,v_0)+\left.\dfrac{\partial g}{\partial x}\right|_{x=x_0,v=v_0}(x-x_0)+\left.\dfrac{\partial g}{\partial v}\right|_{x=x_0,v=v_0}(v-v_0).$$
If $x_0,v_0$ is an equilibrium point we know that $f(x_0,v_0)=g(x_0,v_0)=0$. If we introduce the new state variables $z_1=x-x_0$ and $z_2=v-v_0$ we can rewrite an approximate version of the initial equation as
$$\dot{z}_1=\left.\dfrac{\partial f}{\partial x}\right|_{x=x_0,v=v_0}z_1+\left.\dfrac{\partial f}{\partial v}\right|_{x=x_0,v=v_0}z_2$$ $$\dot{z}_2=\left.\dfrac{\partial g}{\partial x}\right|_{x=x_0,v=v_0}z_1+\left.\dfrac{\partial g}{\partial v}\right|_{x=x_0,v=v_0}z_2.$$
This is a linear system with the system matrix (which is the Jacobian of the nonlinear system) evaluated at $x_0,v_0$.
$$ J=\begin{bmatrix} \dfrac{\partial f}{\partial x} & \dfrac{\partial f}{\partial v}\\ \dfrac{\partial g}{\partial x} & \dfrac{\partial g}{\partial v}\\ \end{bmatrix}_{x=x_0,v=v_0} $$
If you permute the columns or the rows nothing will change algebraically, but the correspondence with the variables $z_1$ and $z_2$ will change.