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<l\le n}(x_l-x_k)^2}{\prod_{1\le,k,l\le n}(x_k+x_l)}. $$

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.