Lower and upper bounds for $H$-eigenvalues of even order real symmetric tensors

TL;DR

This paper introduces new classes of tensors and constructs bounds for their $H$-eigenvalues, providing tighter inclusion sets and extending matrix eigenvalue bounds to tensors.

Contribution

It defines double and quasi-double $ar{B}$-tensors and establishes new eigenvalue inclusion regions for even order symmetric tensors.

Findings

Double $ar{B}$-intervals contain all $H$-eigenvalues.

Quasi-double $ar{B}$-intervals provide additional eigenvalue bounds.

Intervals are nested, refining existing eigenvalue inclusion sets.

Abstract

In this article, we define new classes of tensors called double $\overline{B}$-tensors, quasi-double $\overline{B}$-tensors and establish some of their properties. Using these properties, we construct new regions viz., double $\overline{B}$-intervals and quasi-double $\overline{B}$-intervals, which contain all the $H$-eigenvalues of real even order symmetric tensors. We prove that the double $\overline{B}$-intervals is contained in the quasi-double $\overline{B}$-intervals and quasi-double ${\overline{B}}$-intervals provide supplement information on the Brauer-type eigenvalues inclusion set of tensors. These are analogous to the double $\overline{B}$-intervals of matrices established by J. M. Pe\~na~[On an alternative to Gerschgorin circles and ovals of Cassini, Numer. Math. 95 (2003), no. 2, 337-345.]