二維逆問題/一個無限例子

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. 

練習 (**). 證明圖片中網路的Dirichlet-to-Neumann運算元在自然邊界條件下滿足該方程。

自對偶自對稱無限圖及其對偶

(提示:) 利用運算元/矩陣是Schur補的固定點這一事實：$\Lambda _{G}={\begin{pmatrix}2I&B\\B^{T}&\Lambda +2I\end{pmatrix}}/(\Lambda +2I),$ 其中 $B=-{\begin{pmatrix}1&0&0&\ldots &1\\1&1&0&\ldots &0\\0&\vdots &\ddots &\ddots &\vdots \\\vdots &\vdots &\ddots &1&0\\0&0&\ldots &1&1\\\end{pmatrix}}$ 是迴圈矩陣，使得 $L_{G}=4I-BB^{T}.$