It has been noticed in Landau [9] that the inverse eigenvalue problem is intimately related to orthogonal polynomials. Our treatment in section 2 can also be interpreted that way.
To begin with it is clear that 
, 
are polynomials of degree k-1 in 
. Thus (3.5) simply
states that the polynomials 
, 
, are the
orthonormal polynomials with respect to the scalar product
in the space of polynomials of degree n-1. Writing 
in terms
of the orthogonal polynomials we get
or
with a lower triangular matrix L. This is (3.6). Thus solving the inverse eigenvalue problem, i.e. finding L, boils down to computing the orthogonal polynomials with respect to the scalar product (6.1).
The approach of Gladwell and Willms [7] is
essentially equivalent to our treatment of the inverse eigenvalue
problems, except that we made the role of transmutation more explicit.
For instance, equation (10) of [7] is precisely
our relation (3.6) in the form 
.
