The famous Sylvester matrix was studied in 1854. It is defined as showed in the picture uploaded here. My question is: What and where is the inverse of the Sylvester matrix?
Denote A the given matrix and K the matrix such that K=A-xI where I is a unit matrix. The matrix K is known as Kac, Kac-Sylvester, Sylvester-Kac or Clement matrix.
This information might be useful, although does not give a direct answer to the question.
Dear Profesor Feng Qi, I am sending you some remarks about your very interesting question. Let me know if they present some interest for you. With my best regards