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

Abstract

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

CLC Number:

O151.21

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.

References

[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.