Ghost matrices and a characterization of symmetric Sobolev bilinear forms

In this paper, we characterize symmetric Sobolev bilinear forms defined on P×P, where P is the space of polynomials. More specifically we show that symmetric Sobolev bilinear forms, like symmetric matrices, can be re-written with a diagonal representation. As an application, we introduce the notion of a ghost matrix, extending some classic work of Stieltjes.