# On 2D Inverse Problems/Cauchy matrices

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\le k
It follows, that if $x_k$'s are distinct positive numbers, then the Cauchy matrix 's positive definite.

Exercise (*). Prove that for any positive numbers $x_k$ there is a Stieltjes continued fraction, interpolating the constant unit function at these numbers, $\beta_x(x_k) = 1$.

(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 $\beta_M(M) = \sqrt{M}$.
The next chapter is devoted to the applications of the functional equation.