Upper bounds for Z<inline-formula><tex-math id="M1">\begin{document}$ _1 $\end{document}</tex-math></inline-formula>-eigenvalues of generalized Hilbert tensors

Abstract

The Z 1-eigenvalue of tensors (hypermatrices) was widely used to discuss the properties of higher order Markov chains and transition probability tensors. In this paper, we extend the concept of Z 1-eigenvalue from finite-dimensional tensors to infinite-dimensional tensors, and discuss the upper bound of such eigenvalues for infinite-dimensional generalized Hilbert tensors. Furthermore, an upper bound of Z 1-eigenvalue for finite-dimensional generalized Hilbert tensor is obtained also.

Optimal <i>Z</i>₁-eigenvalue inclusion intervals for tensors and their application

“…Meng and Song [18] introduced the generalized Hilbert tensor as follows: For each positive integer number λ, the entries of an m-order n-dimensional generalized Hilbert tensor H n λ = (H i1i2...im ) are defined by…”

“…Some bounds of Z 1 -spectral radius of tensors can be found in [17,18,23]. The well-known Brauer's eigenvalue inclusion set of matrices was given in [1], which is always contained in the Gershgorin set.…”

“…Examples. In this section, we give two examples to show that our inclusion sets are better than the results in [13,16,18,23].…”