二维逆问题/一个无限例子

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. 

练习 (**). 证明图片中网络的Dirichlet-to-Neumann算子在自然边界条件下满足该方程。

自对偶自对称无限图及其对偶

(提示:) 利用算子/矩阵是Schur补的固定点这一事实：$\Lambda _{G}={\begin{pmatrix}2I&B\\B^{T}&\Lambda +2I\end{pmatrix}}/(\Lambda +2I),$ 其中 $B=-{\begin{pmatrix}1&0&0&\ldots &1\\1&1&0&\ldots &0\\0&\vdots &\ddots &\ddots &\vdots \\\vdots &\vdots &\ddots &1&0\\0&0&\ldots &1&1\\\end{pmatrix}}$ 是循环矩阵，使得 $L_{G}=4I-BB^{T}.$