Thin Hessenberg pairs and double Vandermonde matrices

A square matrix is called Hessenberg whenever each entry below the subdiagonal is zero and each entry on the subdiagonal is nonzero. Let V denote a nonzero finite-dimensional vector space over a field K. We consider an ordered pair of linear transformations A:V→V and A∗:V→V which satisfy both (i), (ii) below.(i)There exists a basis for V with respect to which the matrix representing A is Hessenberg and the matrix representing A∗ is diagonal.(ii)There exists a basis for V with respect to which the matrix representing A is diagonal and the matrix representing A∗ is Hessenberg.We call such a pair a thin Hessenberg pair (or TH pair). By the diameter of the pair we mean the dimension of V minus one. There is an “oriented” version of a TH pair called a TH system. In this paper we investigate a connection between TH systems and double Vandermonde matrices. We have two main results.For the first result we give a bijection between any two of the following three sets:
• The set of isomorphism classes of TH systems over K of diameter d.
• The set of normalized west–south Vandermonde systems in Matd+1(K).
• The set of parameter arrays over K of diameter d.For the second result we give a bijection between any two of the following five sets:
• The set of affine isomorphism classes of TH systems over K of diameter d.
• The set of isomorphism classes of RTH systems over K of diameter d.
• The set of affine classes of normalized west–south Vandermonde systems in Matd+1(K).
• The set of normalized west–south Vandermonde matrices in Matd+1(K).
• The set of reduced parameter arrays over K of diameter d.