A new class of positive semi-definite tensors

Yi Xu; Jinjie Liu; Liqun Qi

doi:10.3934/jimo.2018186

Article Contents

2020, Volume 16, Issue 2: 933-943. Doi: 10.3934/jimo.2018186

This issue Previous Article Extension of generalized solidarity values to interval-valued cooperative games Next Article Convergence analysis of a new iterative algorithm for solving split variational inclusion problems

A new class of positive semi-definite tensors

1.
Mathematics Department, Southeast University, 2 Sipailou, Nanjing, Jiangsu Province 210096, China
2.
Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China

^* Corresponding author: Jinjie Liu
^* Corresponding author: Jinjie Liu

Received: February 28, 2018

Revised: August 31, 2018

Early access: December 2018

Published: February 2020

The first author is supported by National Natural Science Foundation of China Nos. 11501100, 11571178 and 11671082. The third author is supported in part by the Hong Kong Research Grant Council Nos. PolyU 15302114, 15300715, 15301716 and 15300717

Abstract Full Text(HTML) Related Papers Cited by

Abstract

In this paper, a new class of positive semi-definite tensors, the MO tensor, is introduced. It is inspired by the structure of Moler matrix, a class of test matrices. Then we focus on two special cases in the MO-tensors: Sup-MO tensor and essential MO tensor. They are proved to be positive definite tensors. Especially, the smallest H-eigenvalue of a Sup-MO tensor is positive and tends to zero as the dimension tends to infinity, and an essential MO tensor is also a completely positive tensor.

Keywords:

Mathematics Subject Classification: Primary: 15A18, 15A69; Secondary: 15B99.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Springer, New York, 2000. doi: 10.1007/978-1-4612-1394-9.
[2]	H. Chen, G. Li and L. Qi, SOS tensor decomposition: Theory and applications, Commun. Math. Sci., 14 (2016), 2073-2100. doi: 10.4310/CMS.2016.v14.n8.a1.
[3]	A. Cichocki, R. Zdunek, A. H. Phan and S. Amari, Nonnegative Matrix and Tensor Factorizations: Applications to Exploratory Multiway Data Analysis and Blind Source Separation, Wiley, New York, 2009. doi: 10.1002/9780470747278.
[4]	D. Hilbert, Über die Darstellung definiter Formen als Summe von Formenquadraten, Math. Ann., 32 (1888), 342-350. doi: 10.1007/BF01443605.
[5]	C. J. Hillar and L. H. Lim, Most tensor problems are NP-hard, J. ACM, 60 (2013), Art. 45, 39 pp. doi: 10.1145/2512329.
[6]	T. G. Kolda and B. W. Bader, Tensor decompositions and applications, SIAM Rev., 51 (2009), 455-500. doi: 10.1137/07070111X.
[7]	C. Li, F. Wang, J. Zhao, Y. Zhu and Y. Li, Criterions for the positive definiteness of real supersymmetric tensors, J. Comput. Applied. Math., 255 (2014), 1-14. doi: 10.1016/j.cam.2013.04.022.
[8]	Z. Luo and L. Qi, Positive semidefinite tensors (in Chinese), Sci. Sin. Math., 46 (2016), 639-654.
[9]	Z. Luo and L. Qi, Completely positive tensors: Properties, easily checkable subclasses and tractable relaxations, SIAM J. Matrix Anal. Appl., 37 (2016), 1675-1698. doi: 10.1137/15M1025220.
[10]	J. C. Nash, Compact Numerical Methods for Computers: Linear Algebra and Function Minimisation, CRC Press, 1990.
[11]	L. Qi, Eigenvalues of a real supersymmetric tensor, J. Symbolic Comput., 40 (2005), 1302-1324. doi: 10.1016/j.jsc.2005.05.007.
[12]	L. Qi, H$^+$-eigenvalues of Laplacian and signless Laplacian tensors, Commun. Math. Sci., 12 (2014), 1045-1064. doi: 10.4310/CMS.2014.v12.n6.a3.
[13]	L. Qi and Z. Luo, Tensor Analysis: Spectral Theory and Special Tensors, SIAM, Philadelphia, 2017. doi: 10.1137/1.9781611974751.
[14]	L. Qi and Y. Song, An even order symmetric B tensor is positive definite, Linear Algebra Appl., 457 (2014), 303-312. doi: 10.1016/j.laa.2014.05.026.
[15]	L. Qi, C. Xu and Y. Xu, Nonnegative tensor factorization, completely positive tensors, and a hierarchical elimination algorithm, SIAM J Marix Anal. Appl., 35 (2014), 1227-1241. doi: 10.1137/13092232X.
[16]	L. Qi, G. Yu and E. X. Wu, Higher order positive semidefinite diffusion tensor imaging, SIAM J Imaging Sci., 3 (2010), 416-433. doi: 10.1137/090755138.
[17]	L. Zhang, L. Qi and G. Zhou, $M$-tensors and some applications, SIAM J. Matrix Anal. Appl., 35 (2014), 437-452. doi: 10.1137/130915339.