Last modified on 7 May 2015, at 05:06

On 2D Inverse Problems/Cauchy matrices

Let X = {xk} be an ordered set of n complex numbers. The corresponding Cauchy matrix is the matrix CX w/the entries C_X = \{\frac{1}{x_k+x_l}\}.

Principal submatrices of a Cauchy matrix are Cauchy matrices. If x_k's are distinct positive numbers then the Cauchy matrix CX is positive definite.

Exercise (*). Prove that for any positive numbers X = {xk} there is a Stieltjes continued fraction \beta_X interpolating the constant unit function at these numbers, that is \beta_X(x_k) = 1.

(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 \beta_M such that, \beta_M(M) = \sqrt{M}.

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