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}\}

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

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

\det(C_X) = \frac{\prod_{1\le k<l\le n}(x_l-x_k)^2}{\prod_{1\le,k,l\le n}(x_k+x_l)}.

It follows that if the set xk consists of distinct positive numbers then the Cauchy matrix CX is positive definite.

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

\beta_X(x_k) = 1.

(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 \beta_M such that,

\beta_M(M) = \sqrt{M}.

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

Last modified on 18 January 2013, at 00:44