# On 2D Inverse Problems/Cauchy matrices

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

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

${\displaystyle \det(C_{x})={\frac {\prod _{1\leq k
It follows, that if ${\displaystyle x_{k}}$'s are distinct positive numbers, then the Cauchy matrix 's positive definite.

Exercise (*). Prove that for any positive numbers ${\displaystyle x_{k}}$ there is a Stieltjes continued fraction, interpolating the constant unit function at these numbers, ${\displaystyle \beta _{x}(x_{k})=1}$.

(Hint.) Use the solution of the Pick-Nevanlinna interpolation problem w/the appropriate Cauchy matrix.

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

Exercise (**). Prove that for any positive definite matrix M there is a Stieltjes continued fraction, such that ${\displaystyle \beta _{M}(M)={\sqrt {M}}}$.
The next chapter is devoted to the applications of the functional equation.