New Inclusion Sets for the Eigenvalues of Stochastic Tensors

Subject Areas : International Journal of Mathematical Modelling & Computations

Ali Reza Shojaeifard ¹ , Ramin Nasiri ²

1 - Department of Mathematics, Faculty of Sciences, Imam Hossein Comprehensive University, Tehran, Iran.
2 - Department of Mathematics, Faculty of Sciences, Imam Hossein Comprehensive University, Tehran, Iran.

Received: 2021-06-28 Accepted : 2022-01-20 Published : 2022-06-15

Keywords: eigenvalue, Nonnegative tensors, Inclusion sets, Stochastic tensors,

Abstract :

The purpose of this paper is to locate and estimate the eigenvalues of stochastic tensors. We present several estimation theorems about the eigenvalues of stochastic tensors. Meanwhile, we obtain the distribution theorem for the eigenvalues of tensor product of two stochastic tensors. We will conclude the paper with the distribution for the eigenvalues of generalized stochastic tensors.

References:

[1] A. Brauer, Limits for the characteristic roots of a matrix. IV: Applications to stochastic matrices,
Duke Math. J., 19 (1952) 75–91.
[2] C. Bu, X. Jin, H. Li and C. Deng, Brauer-type eigenvalue inclusion sets and the spectral radius of
tensors, Linear Algebra Appl., 512 (2017) 234–248.
[3] C. Li, A. Jiao and Y. Li, An S-type eigenvalue localization set for tensors, Linear Algebra Appl.,
493 (2016) 469–483.
[4] C. Li, J. Zhou and Y. Li, A new Brauer-type eigenvalue localization set for tensors, Linear Multilinear
Algebra, 64 (2016) 727–736.
[5] L. H. Lim, Singular values and eigenvalues of tensors, a variational approach, in: Proceedings
1st IEEE International Workshop on Computational Advances of Multitensor Adaptive Processing,
(2005) 129–132.
[6] L. Qi, Eigenvalues of a real supersymmetric tensor, J. Symbolic Comput., 40 (2005) 1302–1324.
[7] C. Sang and C. Li, Exclusion sets in eigenvalue localization sets for tensors, Linear Multilinear
Algebra, 67 (12) (2019) 2399–2409.
[8] J. Y. Shao, A general product of tensors with applications, Linear Algebra Appl., 439 (2013)
2350–2366.
[9] Y. Yang and Q. Yang, Further results for Perron-Frobenius Theorem for nonnegative tensors II,
Siam J. Matrix Anal. Appl., 32 (2011) 1236–1250.
[10] J. Zhao and C. Sang, An eigenvalue localization set for tensors and its applications, J. Inequal
Appl., 1 (2017) 1–9.