Using the method of lines to the homogenious heat equation $\textbf{u}_t - \textbf{u}_{xx}= 0$ yields the system
$$ \frac{d\textbf{u}}{dt} = A\textbf{u} + \textbf{b}(t) $$
Using ode23s and specifying $A$ as the Jacobian really speeds up the calculations, compared to the regular ode23s.
I've searched some and not found anything, how come the Jacobian for the system is $A$ and why is specifying $A$ as the Jacobian much more efficient?