On 2D Inverse Problems/Cauchy matrices
Let be an ordered set of n complex 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.