# American Institute of Mathematical Sciences

January  2017, 22(1): 187-198. doi: 10.3934/dcdsb.2017009

## Z-Eigenvalue Inclusion Theorems for Tensors

 1 School of Management Science, Qufu Normal University, Rizhao, Shandong 276826, China 2 Department of Mathematics and Statistics, Curtin University, Perth, Australia

* Corresponding author: Guanglu Zhou

Received  December 2015 Revised  May 2016 Published  December 2016

Fund Project: The first author is supported by the natural science foundation of Shandong Province grant ZR2016AM10 and the Fundamental Research Funds for Qufu Normal University grant xkj201415, xkj201314

In this paper, we establish $Z$-eigenvalue inclusion theorems for general tensors, which reveal some crucial differences between $Z$-eigenvalues and $H$-eigenvalues. As an application, we obtain upper bounds for the largest $Z$-eigenvalue of a weakly symmetric nonnegative tensor, which are sharper than existing upper bounds.

Citation: Gang Wang, Guanglu Zhou, Louis Caccetta. Z-Eigenvalue Inclusion Theorems for Tensors. Discrete & Continuous Dynamical Systems - B, 2017, 22 (1) : 187-198. doi: 10.3934/dcdsb.2017009
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##### References:
 $\mathcal{L}_{1,2}(\mathcal{A})=\{\lambda\in C: |\lambda|\leq 4\}$ $\mathcal{L}_{1,3}(\mathcal{A})=\{\lambda\in C: |\lambda|\leq \frac{3+\sqrt{29}}{2}\}$ $\mathcal{L}_{2,1}(\mathcal{A})=\{\lambda\in C: |\lambda|\leq 4\}$ $\mathcal{L}_{2,3}(\mathcal{A})=\{\lambda\in C: |\lambda|\leq 4\}$ $\mathcal{L}_{3,1}(\mathcal{A})=\{\lambda\in C: |\lambda|\leq 2+2\sqrt{2}\}$ $\mathcal{L}_{3,2}(\mathcal{A})=\{\lambda\in C: |\lambda|\leq 5\}$
 $\mathcal{L}_{1,2}(\mathcal{A})=\{\lambda\in C: |\lambda|\leq 4\}$ $\mathcal{L}_{1,3}(\mathcal{A})=\{\lambda\in C: |\lambda|\leq \frac{3+\sqrt{29}}{2}\}$ $\mathcal{L}_{2,1}(\mathcal{A})=\{\lambda\in C: |\lambda|\leq 4\}$ $\mathcal{L}_{2,3}(\mathcal{A})=\{\lambda\in C: |\lambda|\leq 4\}$ $\mathcal{L}_{3,1}(\mathcal{A})=\{\lambda\in C: |\lambda|\leq 2+2\sqrt{2}\}$ $\mathcal{L}_{3,2}(\mathcal{A})=\{\lambda\in C: |\lambda|\leq 5\}$
 $\mathcal{M}_{1,2}(\mathcal{A})=\{\lambda\in C: 3\leq |\lambda|\leq 4\}$ $\mathcal{M}_{1,3}(\mathcal{A})=\{\lambda\in C: |\lambda|\leq \frac{7+\sqrt{5}}{2}\}$ $\mathcal{M}_{2,1}(\mathcal{A})=\{\lambda\in C: 2\leq |\lambda|\leq 4\}$ $\mathcal{M}_{2,3}(\mathcal{A})=\{\lambda\in C: |\lambda|\leq 4\}$ $\mathcal{M}_{3,1}(\mathcal{A})=\{\lambda\in C: 3+\sqrt{3}\leq |\lambda|\leq 5\}$ $\mathcal{M}_{3,2}(\mathcal{A})=\{\lambda\in C: 3\leq |\lambda|\leq 5\}$.
 $\mathcal{M}_{1,2}(\mathcal{A})=\{\lambda\in C: 3\leq |\lambda|\leq 4\}$ $\mathcal{M}_{1,3}(\mathcal{A})=\{\lambda\in C: |\lambda|\leq \frac{7+\sqrt{5}}{2}\}$ $\mathcal{M}_{2,1}(\mathcal{A})=\{\lambda\in C: 2\leq |\lambda|\leq 4\}$ $\mathcal{M}_{2,3}(\mathcal{A})=\{\lambda\in C: |\lambda|\leq 4\}$ $\mathcal{M}_{3,1}(\mathcal{A})=\{\lambda\in C: 3+\sqrt{3}\leq |\lambda|\leq 5\}$ $\mathcal{M}_{3,2}(\mathcal{A})=\{\lambda\in C: 3\leq |\lambda|\leq 5\}$.
 $\mathcal{N}_{1,2}(\mathcal{A})=\{\lambda\in C: |\lambda|\leq \frac{3+\sqrt{21}}{2}\}$ $\mathcal{N}_{1,3}(\mathcal{A})=\{\lambda\in C: |\lambda|\leq \frac{3+\sqrt{29}}{2}$ $\mathcal{N}_{2,1}(\mathcal{A})=\{\lambda\in C: |\lambda|\leq 4\}$ $\mathcal{N}_{2,3}(\mathcal{A})=\{\lambda\in C: |\lambda|\leq \frac{3+\sqrt{29}}{2}$ $\mathcal{N}_{3,1}(\mathcal{A})=\{\lambda\in C: |\lambda|\leq 2+2\sqrt{2}\}$ $\mathcal{N}_{3,2}(\mathcal{A})=\{\lambda\in C: |\lambda|\leq 4\}$.
 $\mathcal{N}_{1,2}(\mathcal{A})=\{\lambda\in C: |\lambda|\leq \frac{3+\sqrt{21}}{2}\}$ $\mathcal{N}_{1,3}(\mathcal{A})=\{\lambda\in C: |\lambda|\leq \frac{3+\sqrt{29}}{2}$ $\mathcal{N}_{2,1}(\mathcal{A})=\{\lambda\in C: |\lambda|\leq 4\}$ $\mathcal{N}_{2,3}(\mathcal{A})=\{\lambda\in C: |\lambda|\leq \frac{3+\sqrt{29}}{2}$ $\mathcal{N}_{3,1}(\mathcal{A})=\{\lambda\in C: |\lambda|\leq 2+2\sqrt{2}\}$ $\mathcal{N}_{3,2}(\mathcal{A})=\{\lambda\in C: |\lambda|\leq 4\}$.
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