I initially asked this question on math.stackexchange.com, but nobody answer so I thought I could try to ask here, this at least to some degree related to electrical engineering.
Please help me to correct my understanding.
$$C^{-1}y+Ax=b \\ A^Ty =f$$
This pair is equilibrium equations, \$C^{-1}+Ax=b\$ represents Ohm's law, derived from:
$$e=b-Ax$$
Vector \$x\$ represent potential on each node in a graph, on each node present abstract force which repel abstract flow. Flow goes to lower potential (which repel less). Act of multiplication \$Ax\$ produces potential difference. I just add all columns vectors adjusted by corresponding potential on each node, this is certainly should give me potential differences on each edge. But formula telling me:
$$Ax=b-e$$
This is my problem. What \$e\$ stands for? I know \$b\$ - potential differences, I can find potential at each node, I know \$x\$, I can find differences. What does \$e\$ mean? In another words if I know potential at each node, then \$C^{-1}y=Ax\$?