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.

Keywords

Tensor / symmetric / Hankel / Vandermonde tensor / generalized Vandermonde tensor

References

[1] Chen H, Chen Z, Qi L. Centrosymmetric, skew centrosymmetric and centrosymmetric Cauchy Tensors. 2014, arXiv: 1406.7409

[2] Chen H, Qi L. Positive definiteness and semi-definiteness of even order symmetric Cauchy tensors. J Ind Manag Optim, 2015, 11: 1263–1274

[3] Comon P, Golub G, Lim L-H, Mourrain B. Symmetric tensors and symmetric tensor rank. SIAM J Matrix Anal Appl, 2008, 30: 1254–1279

[4] Hu S, Huang Z, Ling C, Qi L. On determinants and eigenvalue theory of tensors. J Symbolic Comput, 2011, 50: 508–531

[5] Hu S, Qi L. The E-eigenvectors of tensors. Linear Multlinear Algebra, 2014, 62: 1388–1402

[6] Kaiman D. The generalized Vandermonde matrix. Math Mag, 1984, 57: 15–21

[7] Lim L-H. Singular values and eigenvalues of tensors: A variational approach. In: Proc of 1st IEEE Int’l Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP). 2005, 129–132

[8] Qi L. Eigenvalues of a real supersymmetric tensor. J Symbolic Comput, 2005, 40: 1302–1324

[9] Qi L. Hankel tensors: Associated Hankel matrices and Vandermonde decomposition. Commun Math Sci, 2015, 13: 113–125

[10] Qi L, Xu C, Xu Y. Nonnegative tensor factorization, completely positive tensors and an hierarchically elimination algorithm. 2014, 35: 1227–1241

[11] Shao J, Shan H, Zhang L. On some properties of the determinants of tensors. Linear Algebra Appl, 2013, 439: 3057–3069

[12] Xu C. Hankel tensor, Vandermonde tensors, and their positivities. Linear Algebra Appl, 2016, 491: 56–72

[13] Ycart B. A case of mathematical eponym: the Vandermonde determinant. 2012, arXiv: 1204.4716v1

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag Berlin Heidelberg