# American Institute of Mathematical Sciences

September  2020, 16(5): 2551-2562. doi: 10.3934/jimo.2019069

## Brualdi-type inequalities on the minimum eigenvalue for the Fan product of M-tensors

 School of Management Science, Qufu Normal University, Rizhao, Shandong 276826, China

* Corresponding author: Gang Wang

Received  September 2018 Revised  March 2019 Published  September 2020 Early access  July 2019

Fund Project: This work was supported by the Natural Science Foundation of China (11671228) and the Natural Science Foundation of Shandong Province (ZR2016AM10)

In this paper, we focus on some inequalities for the Fan product of $M$-tensors. Based on Brualdi-type eigenvalue inclusion sets of $M$-tensors and similarity transformation methods, we establish Brualdi-type inequalities on the minimum eigenvalue for the Fan product of two $M$-tensors. Furthermore, we discuss the advantages of different Brualdi-type inequalities. Numerical examples verify the validity of the conclusions.

Citation: Gang Wang, Yiju Wang, Yuan Zhang. Brualdi-type inequalities on the minimum eigenvalue for the Fan product of M-tensors. Journal of Industrial and Management Optimization, 2020, 16 (5) : 2551-2562. doi: 10.3934/jimo.2019069
##### References:

show all references

##### References:
 [1] Chaoqian Li, Yajun Liu, Yaotang Li. Note on $Z$-eigenvalue inclusion theorems for tensors. Journal of Industrial and Management Optimization, 2021, 17 (2) : 687-693. doi: 10.3934/jimo.2019129 [2] Gang Wang, Yuan Zhang. $Z$-eigenvalue exclusion theorems for tensors. Journal of Industrial and Management Optimization, 2020, 16 (4) : 1987-1998. doi: 10.3934/jimo.2019039 [3] Caili Sang, Zhen Chen. $E$-eigenvalue localization sets for tensors. Journal of Industrial and Management Optimization, 2020, 16 (4) : 2045-2063. doi: 10.3934/jimo.2019042 [4] Juan Meng, Yisheng Song. Upper bounds for Z$_1$-eigenvalues of generalized Hilbert tensors. Journal of Industrial and Management Optimization, 2020, 16 (2) : 911-918. doi: 10.3934/jimo.2018184 [5] Caili Sang, Zhen Chen. Optimal $Z$-eigenvalue inclusion intervals of tensors and their applications. Journal of Industrial and Management Optimization, 2022, 18 (4) : 2435-2468. doi: 10.3934/jimo.2021075 [6] Liyuan Tian, Yong Wang. Solving tensor complementarity problems with $Z$-tensors via a weighted fixed point method. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022093 [7] Zalman Balanov, Yakov Krasnov. On good deformations of $A_m$-singularities. Discrete and Continuous Dynamical Systems - S, 2019, 12 (7) : 1851-1866. doi: 10.3934/dcdss.2019122 [8] Jennifer D. Key, Bernardo G. Rodrigues. Binary codes from $m$-ary $n$-cubes $Q^m_n$. Advances in Mathematics of Communications, 2021, 15 (3) : 507-524. doi: 10.3934/amc.2020079 [9] Chaoqian Li, Yaqiang Wang, Jieyi Yi, Yaotang Li. Bounds for the spectral radius of nonnegative tensors. Journal of Industrial and Management Optimization, 2016, 12 (3) : 975-990. doi: 10.3934/jimo.2016.12.975 [10] Yuyan Yao, Gang Wang. Sharp upper bounds on the maximum $M$-eigenvalue of fourth-order partially symmetric nonnegative tensors. Mathematical Foundations of Computing, 2022, 5 (1) : 33-44. doi: 10.3934/mfc.2021018 [11] Yonglin Cao, Yuan Cao, Hai Q. Dinh, Fang-Wei Fu, Jian Gao, Songsak Sriboonchitta. Constacyclic codes of length $np^s$ over $\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}$. Advances in Mathematics of Communications, 2018, 12 (2) : 231-262. doi: 10.3934/amc.2018016 [12] Roghayeh Mohammadi Hesari, Mahboubeh Hosseinabadi, Rashid Rezaei, Karim Samei. $\mathbb{F}_{p^{m}}\mathbb{F}_{p^{m}}{[u^2]}$-additive skew cyclic codes of length $2p^s$. Advances in Mathematics of Communications, 2022  doi: 10.3934/amc.2022023 [13] Hai Q. Dinh, Bac T. Nguyen, Paravee Maneejuk. Constacyclic codes of length $8p^s$ over $\mathbb F_{p^m} + u\mathbb F_{p^m}$. Advances in Mathematics of Communications, 2022, 16 (3) : 525-570. doi: 10.3934/amc.2020123 [14] Hideaki Takagi. Times until service completion and abandonment in an M/M/$m$ preemptive-resume LCFS queue with impatient customers. Journal of Industrial and Management Optimization, 2018, 14 (4) : 1701-1726. doi: 10.3934/jimo.2018028 [15] Jin Wang, Jun-E Feng, Hua-Lin Huang. Solvability of the matrix equation $AX^{2} = B$ with semi-tensor product. Electronic Research Archive, 2021, 29 (3) : 2249-2267. doi: 10.3934/era.2020114 [16] Bing Sun, Liangyun Chen, Yan Cao. On the universal $\alpha$-central extensions of the semi-direct product of Hom-preLie algebras. Electronic Research Archive, 2021, 29 (4) : 2619-2636. doi: 10.3934/era.2021004 [17] Yining Gu, Wei Wu. Partially symmetric nonnegative rectangular tensors and copositive rectangular tensors. Journal of Industrial and Management Optimization, 2019, 15 (2) : 775-789. doi: 10.3934/jimo.2018070 [18] Van Hoang Nguyen. A simple proof of the Adams type inequalities in ${\mathbb R}^{2m}$. Discrete and Continuous Dynamical Systems, 2020, 40 (10) : 5755-5764. doi: 10.3934/dcds.2020244 [19] Chandra Shekhar, Amit Kumar, Shreekant Varshney, Sherif Ibrahim Ammar. $\bf{M/G/1}$ fault-tolerant machining system with imperfection. Journal of Industrial and Management Optimization, 2021, 17 (1) : 1-28. doi: 10.3934/jimo.2019096 [20] Habibul Islam, Om Prakash, Ram Krishna Verma. New quantum codes from constacyclic codes over the ring $R_{k,m}$. Advances in Mathematics of Communications, 2022, 16 (1) : 17-35. doi: 10.3934/amc.2020097