On 2D Inverse Problems/An infinite example

The following construction provides an example of an infinite network (featured on the cover of the book), which Dirichlet-to-Neumann operator satisfies the equation in the title of this chapter. 

$\Lambda _{G}={\sqrt {L}}.$

The matrix equation reflects the self-duality and self-symmetry of the network. 

Exercise (**). Prove that the Dirichlet-to-Neumann operator of the network on the picture w/the natural boundary satisfies the equation.

The self-dual self-symmetric infinite graph w/its dual

(Hint:) Use the fact that the operator/matrix is the fixed point of the Schur complement]]: $\Lambda _{G}={\begin{pmatrix}2I&B\\B^{T}&\Lambda +2I\end{pmatrix}}/(\Lambda +2I),$ where $B=-{\begin{pmatrix}1&0&0&\ldots &1\\1&1&0&\ldots &0\\0&\vdots &\ddots &\ddots &\vdots \\\vdots &\vdots &\ddots &1&0\\0&0&\ldots &1&1\\\end{pmatrix}}$ is the circulant matrix, such that $L_{G}=4I-BB^{T}.$