# On 2D Inverse Problems/Cauchy matrices

Let X = {xk} be an ordered set of n complex numbers. The corresponding Cauchy matrix is the matrix CX w/the entries $C_X = \{\frac{1}{x_k+x_l}\}$.

Principal submatrices of a Cauchy matrix are Cauchy matrices. If $x_k$'s are distinct positive numbers then the Cauchy matrix CX is positive definite.

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

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

The above exercise has the following functional equation corollary, that's true for the discrete and continuous Dirichlet-to-Neumann maps.

Exercise (**). Prove that for any positive definite matrix M there is a Stieltjes continued fraction $\beta_M$ such that, $\beta_M(M) = \sqrt{M}$.

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