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.

(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 $$L_G = 4I-BB^T.$$