User:DVD206/Effective resistances and Dirichlet-to-Neumann operator

The Dirichlet-to-Neumann operator is a special type of Poincaré–Steklov operator. It is the pseudo-differential operator from the Dirichlet boundary data to the Neumann boundary data of harmonic functions. It is well-defined because of uniqueness and existence of the solution of the Dirichlet problem.

Let

$M=\left[{\begin{matrix}A&B\\C&D\end{matrix}}\right]$

so that M is a (p+q)×(p+q) matrix.

Then the Schur complement of the block D of the matrix M is the p×p matrix

$\Lambda (G)=A-BD^{-1}C.\,$

The Laplace equation gives the connection between the hitting probability of the random walk started at the boundary and the value of a harmonic function at a point. The connection can be expressed using the sum of the geometric series identity applied to the blocks of the Kirchhoff matrix of the network/graph.

$\Lambda (G)=D(I-H(G))=A-B^{T}\sum (I-C)^{n}B.$

This is a special case of the Neumann series applied to the diagonally dominated matrix.