Consider an equation of the form $$\ddot{x}_i(t) = A*|y_N(t)-y_i(t)|^P+B*|x_E(t)-x_i(t)|^P+C*|y_S(t)-y_i(t)|^P+D*|x_W(t)-x_i(t)|^P+K,A,B,C,D,K,P \in \mathbb{R}$$.
I wish to represent this in the form $$\ddot{x}_i(t) = x_i(t)*AA + x_E(t)*BB+x_W(t)*CC + y_i(t)*DD+y_N(t)*EE+y_S(t)*FF+K$$
so that I can eventually represent equations where $i = 1,2,3,\dots,M$ in a matrix notation. (Note that N, E, S, and W do depend on i so they will go into matrix form nicely). The portion that I am struggling with is representing the absolute values raised to the power P (Note that $P>0.0$). in a form that will translate to a matrix representation well.
Any help is appreciated since I cannot think of a way! I hope that the constants are not confusing -- they represent many other constants but all simply are real numbers.
I do believe this is impossible. I was searching for an obscure trick that I did not know. Upon further research I have to conclude that there is no way to represent a system of nonlinear equations using a matrix -- that is why they call is linear algebra, huh!?