Generalized Vandermonde tensors

摘要

We extend Vandermonde matrices to generalized Vandermonde tensors. We call an mth order n-dimensional real tensor A = (A(i1i2...im)) a type-1 generalized Vandermonde (GV) tensor, or GV(1) tensor, if there exists a vector v = (v(1), v(2), ..., v(n))(T) such that A(i1i2...im) = v(i1)(i2+i3+ ...+im-m+1), and call A a type-2 (mth order n dimensional) GV tensor, or GV(2) tensor, if there exists an (m - 1) th order tensor B = (Bi1i2...im-1) such that A(i1i2...im) = B-i1i2...im-1(im-1). In this paper, we mainly investigate the type-1 GV tensors including their products, their spectra, and their positivities. Applications of GV tensors are also introduced.