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. 

${\mathrm{\Lambda }}_G=\sqrt{L}.$

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]]: ${\mathrm{\Lambda }}_G=\left(\begin{array}{cc}2I& B\\ B^T& \mathrm{\Lambda }+2I\end{array}\right)/(\mathrm{\Lambda }+2I),$ where $B=-\left(\begin{array}{ccccc}1& 0& 0& \dots & 1\\ 1& 1& 0& \dots & 0\\ 0& ⋮& \ddots & \ddots & ⋮\\ ⋮& ⋮& \ddots & 1& 0\\ 0& 0& \dots & 1& 1\end{array}\right)$ is the circulant matrix, such that $L_G=4I-B{B}^T.$