- Let
be a set of n positive numbers. The Stieltjes continued fraction is an expression of the form

The function and its reciprocal define all rational n-to-1 maps of the right half of the complex plane onto itself,
![{\displaystyle \beta ,1/\beta :\mathbb {C^{+}} {\xrightarrow[{}]{n\leftrightarrow 1}}\mathbb {C^{+}} ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9b925058099da9458025e280e32eb7e44cc25033)
since

- The function
is determined by the pre-image set
of the point {z = 1}, since

and a complex polynomial is determined by its roots up to a multiplicative constant by the fundamental theorem of algebra.
Let
be the elementary symmetric functions of the set
. That is,

- Then the coefficients
of the continued fraction are the pivots in the Gauss-Jordan elimination algorithm of the following n by n square Hurwitz matrix:

and, therefore, can be expressed as the ratios of monomials of the determinants of its blocks.
Exercise 1 : Prove that

Exercise 2 :
Let A be a diagonal matrix with the alternating in sign diagonal entries:

and D the (0,1)-matrix

Prove that the continued fraction
evaluated at a point
equals to 1 if and only if
is an eigenvalue of the matrix AD.
Exercise 3 :
Use Exercise 1 to prove that
