# On 2D Inverse Problems/Cauchy matrices

From Wikibooks, open books for an open world

Let be an ordered set ofncomplex numbers. The corresponding Cauchy matrix is the matrix .

Principal submatrices of a Cauchy matrix are Cauchy matrices. The determinant of a Cauchy matrix is given by the following formula:

It follows, that if 's are distinct positive numbers, then the Cauchy matrix 's positive definite.

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

(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 .

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