二維逆問題/柯西矩陣

Let $x_{k}$ be an ordered set of n complex numbers. The corresponding Cauchy matrix is the matrix $C_{x}=\{{\frac {1}{x_{k}+x_{l}}}\}$.

Principal submatrices of a Cauchy matrix are Cauchy matrices. 
The determinant of a Cauchy matrix is given by the following formula:

$\det(C_{x})={\frac {\prod _{1\leq k<l\leq n}(x_{l}-x_{k})^{2}}{\prod _{1\leq ,k,l\leq n}(x_{k}+x_{l})}}.$

It follows, that if $x_{k}$'s are distinct positive numbers, then the Cauchy matrix 's positive definite.

練習 (*)。 證明對於任何正數 $x_{k}$，存在一個 Stieltjes 連分數，在這些數上插值常數單位函式， $\beta _{x}(x_{k})=1$.

(提示。) 使用 Pick-Nevanlinna 插值問題的解，並使用適當的柯西矩陣。

The latter exercise has the following functional equation corollary for the discrete and continuous Dirichlet-to-Neumann maps.

練習 (**) 。 證明對於任何正定矩陣 M，存在一個 Stieltjes 連分數，使得 $\beta _{M}(M)={\sqrt {M}}$.

The next chapter is devoted to the applications of the functional equation.