Generalized Vandermonde tensors

Abstract

We extend Vandermonde matrices to generalized Vandermonde tensors. We call an mth order n-dimensional real tensor A=({{\displaystyle A}}_{{{\displaystyle i}}_{\mathrm{1}}{{\displaystyle i}}_{\mathrm{2}}\mathrm{...}{{\displaystyle i}}_m}) a type-1 generalized Vandermonde (GV) tensor, or GV1 tensor, if there exists a vector v={({{\displaystyle v}}_{\mathrm{1}},{{\displaystyle v}}_{\mathrm{2}}\mathrm{...}{{\displaystyle v}}_n)}^{\mathrm{T}} such that {{\displaystyle A}}_{{{\displaystyle i}}_{\mathrm{1}}{{\displaystyle i}}_{\mathrm{2}}\mathrm{...}{{\displaystyle i}}_m}={{\displaystyle v}}_{{{\displaystyle i}}_{\mathrm{1}}}^{{{\displaystyle i}}_{\mathrm{2}}+{{\displaystyle i}}_{\mathrm{3}}+\mathrm{...}+{{\displaystyle i}}_m-m+\mathrm{1}}, and call A a type-2 (mth order ndimensional) GV tensor, or GV2 tensor, if there exists an (m-1)th order tensor B=({{\displaystyle B}}_{{{\displaystyle i}}_{\mathrm{1}}{{\displaystyle i}}_{\mathrm{2}}\mathrm{...}{{\displaystyle i}}_{m-\mathrm{1}}}) such that {{\displaystyle A}}_{{{\displaystyle i}}_{\mathrm{1}}{{\displaystyle i}}_{\mathrm{2}}\mathrm{...}{{\displaystyle i}}_m}={{\displaystyle B}}_{{{\displaystyle i}}_{\mathrm{1}}{{\displaystyle i}}_{\mathrm{2}}\mathrm{...}{{\displaystyle i}}_{m-\mathrm{1}}}^{{{\displaystyle i}}_m-\mathrm{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.

Key words： Tensor; symmetric; Hankel; Vandermonde tensor; generalized Vandermonde tensor

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