On Lec 2 | MIT 18.085 Computational Science and Engineering I, Professor Strang introduces the stiffness matrix $A^\top C A$ such that in the vertical system of springs in the picture
the following balance equation holds:
$$f = A^\top C A u\tag 1$$
where $f$ is the force of gravity; $u$ is the displacement of the masses at equilibrium; $C$ is the spring stiffness matrix in Hooke's law; and $A,$ my problem:
In $Au$ the displacement of the masses, $u_1,u_2, u_3$ results in different elongation of the springs $e_1,e_2,e_3,e_4, $ encoded in the matrix of first differences (or incidence matrix in a graph or network), $A:$
$$A=\begin{bmatrix} 1&0&0\\ -1&1&0\\ 0&-1&1\\ 0&0&-1 \end{bmatrix}$$
encoding the equations
$$\begin{align} e_1&=u_1\\ e_2&=u_2-u_1\\ e_3&=u_3-u_2\\ e_4&=-u_3 \end{align}$$
On the other hand, the $\underset{-------------}{\text{(fill in the blank)}}$ matrix $A^\top$
$$A^\top=\begin{bmatrix} 1&-1&0&0\\ 0&1&-1&0\\ 0&0&1&-1 \end{bmatrix}$$
encodes
$$\begin{align} w_1-w_2&=m_1g\\ w_2-w_3&=m_2g\\ w_3-w_4&=m_3g\\ \end{align}$$
and
$$\begin{align} w_1&=c_1e_1\\ w_2&=c_2e_2\\ w_3&=c_3e_3\\ w_4&=c_4e_4 \end{align}$$
i.e. the force $w$ is proportional to the length, Hooke's law.
I need to understand in English what $A$ and $A^\top$ are. Do they have generic names?
Evidently, Eq. 1 is not an ad hoc solution to a spring problem, but a rather general model to solve electrical engineering and network problems. So there has to be a more verbal and conceptual explanation in English than A an A transpose.
Right off the bat, $A$ has $4$ rows corresponding to springs (edges in a graph), while $A^\top$ has only $3$ rows corresponding to each mass (nodes in a graph). So I would start thinking of $A$ as a transformation $\left[\text{springs}\right]\left[\text{masses}\right]=\left[\text{springs}\right]$ - i.e. a transformation of the vector of displacement of masses $\mathbf u$ into a vector of elongation of the springs $\mathbf e.$
On the other hand, $A^\top$ would transform the vector of energy in the springs $\mathbf w$ into the vector of gravity force at balance $\mathbf f$ for each mass.
So in a way, we start with displacement of the masses, and we end up with energy at equilibrium of the masses, going through the intermediate step of seeing the effect of the links (springs or nodes) on these masses. It would be the natural way of going from displacement to energy.
But how can this be conceptually generalized?