On the M-eigenvalue estimation of fourth-order partially symmetric tensors

Abstract

In this article, the M-eigenvalue of fourth-order partially symmetric tensors is estimated by choosing different components of M-eigenvector. As an application, some upper bounds for the M-spectral radius of nonnegative fourth-order partially symmetric tensors are discussed, which are sharper than existing upper bounds. Finally, numerical examples are reported to verify the obtained results.

M-Eigenvalues-Based Sufficient Conditions for the Positive Definiteness of Fourth-Order Partially Symmetric Tensors

“…Radius. Based on eorems 1 and 2, we present sharp bound estimations on M-spectral radius of fourth-order partially symmetric nonnegative tensors, which improves the corresponding results in [19,20]. For the medium-sized tensors, we show the validity of the estimations given by our theorems.…”

“…All authors contributed equally to this manuscript. Table 4 References Interval eorem 3.1 of [19] ρ M (A) ≤ 24 eorem 3.3 of [19] ρ M (A) ≤ 24 eorem 3.5 of [19] ρ M (A) ≤ 24 eorem 1 of [20] ρ M (A) ≤ 26 eorem 2 of [20] ρ M (A) ≤ 24 Lemma 4 and eorem 5 11.75 ≤ ρ M (A) ≤ 24 Lemma 4 and eorem 6 11.75 ≤ ρ M (A) ≤ 23.6941…”

“…us, some researchers turned to investigating the inclusion sets of M-eigenvalue [19][20][21]. For example, Che et al [19] proposed a Gershgorin-type Minclusion set as follows.…”

“…It is clear that eorems 1 and 2 reduce to eorems 2.1 and 2.2 of[19] if one takes α � 0, respectively. Consequently, the upper bounds of ρ M (A) in eorems 1 and 2 are smaller than those in eorems 2.1 and 2.2 of[19]. (ii) By using the equation Axyx � λy, we can establish some conclusions similar to eorems 1 and 2.…”

New M-eigenvalue intervals and application to the strong ellipticity of fourth-order partially symmetric tensors

“…The strong ellipticity condition plays important roles in nonlinear elasticity and materials. Furhtermore, the strong ellipticity has received much attention in practice [1,2,19,8,11,14,27], and some necessary and/or sufficient conditions are proposed [13,15,21,22,16].…”