Generalized Vandermonde tensors

doi:10.1007/s11464-016-0528-9

Front. Math. China

2016, Vol. 11

Issue (3) : 593-603 https://doi.org/10.1007/s11464-016-0528-9

RESEARCH ARTICLE

Generalized Vandermonde tensors

Changqing XU(

),Mingyue WANG,Xian LI

School of Mathematics and Physics, Suzhou University of Science and Technology, Suzhou 215009, China

Download: PDF(135 KB)
Export: BibTeX | EndNote | Reference Manager | ProCite | RefWorks

Abstract

We extend Vandermonde matrices to generalized Vandermonde tensors. We call an mth order n-dimensional real tensor A=(Ai1i2...im) a type-1 generalized Vandermonde (GV) tensor, or GV1 tensor, if there exists a vector v=(v1,v2...vn)T such that Ai1i2...im=vi1i2+i3+...+im-m+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=(Bi1i2...im-1) such that Ai1i2...im=Bi1i2...im-1im-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

Corresponding Author(s): Changqing XU

Issue Date: 17 May 2016

Cite this article:

Changqing XU,Mingyue WANG,Xian LI. Generalized Vandermonde tensors[J]. Front. Math. China, 2016, 11(3): 593-603.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-016-0528-9
https://academic.hep.com.cn/fmc/EN/Y2016/V11/I3/593

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 https://doi.org/10.3934/jimo.2015.11.1263
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 https://doi.org/10.1137/060661569
4	Hu S, Huang Z, Ling C, Qi L. On determinants and eigenvalue theory of tensors. J Symbolic Comput, 2011, 50: 508–531 https://doi.org/10.1016/j.jsc.2012.10.001
5	Hu S, Qi L. The E-eigenvectors of tensors. Linear Multlinear Algebra, 2014, 62: 1388–1402 https://doi.org/10.1080/03081087.2013.828721
6	Kaiman D. The generalized Vandermonde matrix. Math Mag, 1984, 57: 15–21 https://doi.org/10.2307/2690290
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 https://doi.org/10.1016/j.jsc.2005.05.007
9	Qi L. Hankel tensors: Associated Hankel matrices and Vandermonde decomposition. Commun Math Sci, 2015, 13: 113–125 https://doi.org/10.4310/CMS.2015.v13.n1.a6
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 https://doi.org/10.1016/j.laa.2013.08.014
12	Xu C. Hankel tensor, Vandermonde tensors, and their positivities. Linear Algebra Appl, 2016, 491: 56–72 https://doi.org/10.1016/j.laa.2015.02.012
13	Ycart B. A case of mathematical eponym: the Vandermonde determinant. 2012, arXiv: 1204.4716v1

[1]	Saeed RAHMATI, Mohamed A. TAWHID. On intervals and sets of hypermatrices (tensors)[J]. Front. Math. China, 2020, 15(6): 1175-1188.
[2]	Hui LIU, Hui ZHANG. Multiple P-cyclic symmetric closed characteristics on compact convex P-cyclic symmetric hypersurfaces in $ℝ 2 n$ [J]. Front. Math. China, 2020, 15(6): 1155-1173.
[3]	Mengyan XIE, Qing-Wen WANG. Reducible solution to a quaternion tensor equation[J]. Front. Math. China, 2020, 15(5): 1047-1070.
[4]	Yizheng FAN, Zhu ZHU, Yi WANG. Least H-eigenvalue of adjacency tensor of hypergraphs with cut vertices[J]. Front. Math. China, 2020, 15(3): 451-465.
[5]	Hongmei YAO, Li MA, Chunmeng LIU, Changjiang BU. Brualdi-type inclusion sets of Z-eigenvalues and l^k,s-singular values for tensors[J]. Front. Math. China, 2020, 15(3): 601-612.
[6]	Gang WANG, Yuan ZHANG, YijuWANG WANG. Brauer-type bounds for Hadamard product of nonnegative tensors[J]. Front. Math. China, 2020, 15(3): 555-570.
[7]	Ziyan LUO, Liqun QI, Philippe L. TOINT. Tensor Bernstein concentration inequalities with an application to sample estimators for high-order moments[J]. Front. Math. China, 2020, 15(2): 367-384.
[8]	Haibin CHEN, Yiju WANG, Guanglu ZHOU. High-order sum-of-squares structured tensors: theory and applications[J]. Front. Math. China, 2020, 15(2): 255-284.
[9]	Lihua YOU, Xiaohua HUANG, Xiying YUAN. Sharp bounds for spectral radius of nonnegative weakly irreducible tensors[J]. Front. Math. China, 2019, 14(5): 989-1015.
[10]	Qingzhi YANG, Yiyong LI. Standard tensor and its applications in problem of singular values of tensors[J]. Front. Math. China, 2019, 14(5): 967-987.
[11]	Neng-Yi WANG. Convergence rates of symmetric scan Gibbs sampler[J]. Front. Math. China, 2019, 14(5): 941-955.
[12]	Zhenyuan NI, Jiancai SUN. Super-biderivations on twisted N = 1 Schrödinger-Neveu-Schwarz algebra[J]. Front. Math. China, 2019, 14(5): 907-922.
[13]	Dong LIU, Xiufu ZHANG. Tensor product weight modules of Schrödinger-Virasoro algebras[J]. Front. Math. China, 2019, 14(2): 381-393.
[14]	Lubin Cui, Minghui Li. Jordan canonical form of three-way tensor with multilinear rank (4,4,3)[J]. Front. Math. China, 2019, 14(2): 281-300.
[15]	Guimei ZHANG, Shenglong HU. Characteristic polynomial and higher order traces of third order three dimensional tensors[J]. Front. Math. China, 2019, 14(1): 225-237.

Viewed

Full text

Abstract

Cited

Shared

Discussed