# 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}Every principal submatrix of a Cauchy matrix is itself a Cauchy matrix.

The determinant of a Cauchy matrix is given by the following formula:

It follows that if the set *x _{k}* consists of distinct positive numbers then the Cauchy matrix

*C*is positive definite.

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

*1*at these numbers, that is

(Hint.) Use the criteria of the existence in the Pick-Nevanlinna interpolation problem w/an appropriate Cauchy matrix.

The above exercise has the following corollary, connecting functional equations for the discrete and continuous Dirichlet-to-Neumann operators.

**Exercise (**).** Prove that for any positive definite matrix *M* there is a Stieltjes continued fraction such that,

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