# On 2D Inverse Problems/Cauchy matrices

Let *X = {x _{k}}* be an ordered set of

*n*complex numbers. The corresponding Cauchy matrix is the matrix

*C*w/the entries .

_{X}Principal submatrices of a Cauchy matrix are Cauchy matrices. If 's are distinct positive numbers then the Cauchy matrix *C _{X}* is positive definite.

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

(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 such that, .

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