四阶弱对称非负张量Z-谱半径的上下界及应用

兰州理工大学学报 ›› 2021, Vol. 47 ›› Issue (3): 162-166.

四阶弱对称非负张量Z-谱半径的上下界及应用

雷学红^*, 许云霞

凯里学院理学院, 贵州凯里 556011

收稿日期:2020-04-24 出版日期:2021-06-28 发布日期:2021-07-19
通讯作者: 雷学红(1978-),男,河南固始人,硕士,讲师.Email: leixuehong123@163.com
基金资助:
国家自然科学基金 (11501141),贵州省教育厅青年科技人才成长项目(黔教合KY字[2019]186号,黔教合KY字[2019]189号)

Upper and lower bounds for the Z-spectral radius of fourth-order weakly symmetric nonnegative tensors and their applications

LEI Xue-hong, XU Yun-xia

College of Science, Kaili University, Kaili 556011, China

Received:2020-04-24 Online:2021-06-28 Published:2021-07-19

摘要/Abstract

摘要： 针对四阶张量Z-谱半径的估计问题,利用张量Z-特征值的定义,并结合不等式放缩技巧,给出了四阶弱对称非负张量Z-谱半径的新上下界,改进了现有一些结果.作为应用,由Z-谱半径的上界给出了张量最佳秩一逼近和贪婪秩一更新算法收敛速度的下界,由Z-谱半径的上下界给出了具有非负振幅对称纯态纠缠的几何度量的上下界.

关键词: 四阶张量, Z-特征值, Z-谱半径, 最佳秩一逼近, 量子纠缠

Abstract: For the bounds of fourth-order tensors, by using the definition of Z-eigenvalues of tensors and some techniques of inequalities, new upper and lower bounds for the Z-spectral radius of fourth-order weakly symmetric nonnegative tensors are obtained and proved to be an improvement of some existing result. As applications, new lower bounds for the best rank-one, approximation of tensors and the convergence rate of the greedy rank-one, update algorithm are given, and lower and upper bounds for symmetric pure state with nonnegative amplitudes are obtained by using upper and lower bounds of the Z-spectral radius of tensors.

Key words: fourth-order tensors, Z-eigenvalues, Z-spectral radius, best rank-one approximation, quantum entanglement

中图分类号:

O151.21

雷学红, 许云霞. 四阶弱对称非负张量Z-谱半径的上下界及应用[J]. 兰州理工大学学报, 2021, 47(3): 162-166.

LEI Xue-hong, XU Yun-xia. Upper and lower bounds for the Z-spectral radius of fourth-order weakly symmetric nonnegative tensors and their applications[J]. Journal of Lanzhou University of Technology, 2021, 47(3): 162-166.

参考文献

[1] KOFIDIS E,REGALIA P A.On the best rank-1 approximation of higher-order supersymmetric tensors [J].SIAM Journal on Matrix Analysis and Applications,2002,23:863-884.
[2] QI L Q.Rank and eigenvalues of a supersymmetric tensor,the multivariate homogeneous polynomial and the algebraic hypersurface itdefines [J].Journal of Symbolic Computation,2006,41:1309-1327.
[3] AMMAR A,CHINEATA F,FALCO A.On the convergence of a greedy rank-one update algorithm for a class of linear systems [J].Archives of Computational Methods in Engineering,2010,17:473-486.
[4] QI L Q.The best rank-one approximation ratio of a tensor space [J].SIAM Journal on Matrix Analysis and Applications,2011,32(2):430-442.
[5] QI L Q,ZHANG G F,Ni G Y.How entangled can a multi-party system possibly be [J].Physics Letters A,2018,382(22):1465-1471.
[6] XIONG L,LIU J Z.Z-eigenvalue inclusion theorem of tensors and the geometric measure of entanglement of multipartite pure states [J].Computational and Applied Mathematics,2020,39(2):13501-13511.
[7] QI L Q.Eigenvalues of a real supersymmetric tensor [J].Journal of Symbolic Computation,2005,40:1302-1324.
[8] CHANG K C,PEARSON K J,ZHANG T.Some variational principles for Z-eigenvalues of nonnegative tensors [J].Linear Algebra and its Applications,2013,438:4166-4182.
[9] WANG G,ZHANG Y.Z-eigenvalue exclusion theorems for tensors [J].Journal of Industrial and Management Optimization,2020,16(4):1987-1998.
[10] HE J,LIU Y M,XU G J.Z-eigenvalues-based sufficient conditions for the positive definiteness of fourth-order tensors[J].Bulletin of the Malaysian Mathematical Sciences Society,2020,43(2):1069-1093.
[11] SANG C L,CHEN Z.E-eigenvalue localization sets for tensors [J].Journal of Industrial and Management Optimization,2020,16(4):2045-2063.
[12] SONG Y S,QI L Q.Spectral properties of positively homogeneous operators induced by higher order tensors [J].SIAM Journal on Matrix Analysis and Applications,2013,34(4):1581-1595.
[13] WANG G,ZHOU G L,CACCETTA L.Z-eigenvalue inclusion theorems for tensors [J].Discrete and Continuous Dynamical Systems Series B,2017,22(1):187-198.
[14] ZHAO J X.E-eigenvalue localization sets for fourth-order tensors [J].Bulletin of the Malaysian Mathematical Sciences Society,2020,43(2):1685-1707.
[15] LI W,LIU D D,VONG S W.Z-eigenpair bounds for an irreducible nonnegative tensor [J].Linear Algebra and its Applications,2015,483:182-199.
[16] ZHAO J X,SANG C L.Two new eigenvalue localization sets for tensors and theirs applications [J].Open Mathematics,2017,15:1267-1276.
[17] WANG Y N,WANG G.Two S-type Z-eigenvalue inclusion sets for tensors [J].Journal of Inequalities and Applications,2017,2017:15201-15212.
[18] HE J.Bounds for the largest eigenvalue of nonnegative tensors [J].Journal of Computational Analysis and Applications,2016,20(7):1290-1301.
[19] HE J,LIU Y M,KE H,et al.Bounds for the Z-spectral radius of nonnegative tensors [J].Springer Plus,2016,5:17271-17278.
[20] LIU Q L,LI Y T.Bounds for the Z-eigenpair of general nonnegative tensors [J].Open Mathematics,2016,14:181-194.
[21] SANG C L.A new Brauer-type Z-eigenvalue inclusion set for tensors [J].Numerical Algorithms,2019,80:781-794.
[22] ZHAO J X.A new Z-eigenvalue localization set for tensors [J].Journal of Inequalities and Applications,2017,2017:851-859.