SHARP BOUNDS ON THE MINIMUM <i>M</i>-EIGENVALUE OF ELASTICITY <i>Z</i>-TENSORS

Sharp bounds on the minimum $M$-eigenvalue and strong ellipticity condition of elasticity $Z$-tensors-tensors

“…Based on the minimum diagonal entries, He et al [10] proposed some bounds for the minimum M -eigenvalue of elasticity M -tensors under irreducible conditions. Combining the maximum diagonal entries with accurate eigenvector information, Wang et al [22] established sharp bound estimations on the minimum M -eigenvalue of elasticity Z-tensors without irreducible conditions, and gave the checkable sufficient conditions for the strong ellipticity condition. It is noted that the bound estimations on the minimum M -eigenvalue of elasticity Z-tensors(or elasticity M -tensors) are established based on elementary operations of tensor elements [10,22].…”

“…Combining the maximum diagonal entries with accurate eigenvector information, Wang et al [22] established sharp bound estimations on the minimum M -eigenvalue of elasticity Z-tensors without irreducible conditions, and gave the checkable sufficient conditions for the strong ellipticity condition. It is noted that the bound estimations on the minimum M -eigenvalue of elasticity Z-tensors(or elasticity M -tensors) are established based on elementary operations of tensor elements [10,22]. To the best of our knowledge, properties of fourth-order tensors are closely related to matrices, and the extreme eigenvalues of the symmetric matrices have a mature calculation technology.…”

“…Recently, Wang et al [22] proposed new characterizations of the minimum Meigenvalue and corresponding to M -eigenvectors for elasticity Z-tensors.…”

“…Lemma 2.4. (Lemma 2.4 of [22]) Let A = (a ijkl ) ∈ E 4,n be an elasticity Z-tensor. Then, there exist nonnegative left and right M -eigenvectors x and y corresponding to τ M (A) such that…”

“…Next, we use Example 4.1 of [22] to show the superiority of our results. As we know, the strong ellipticity condition holds if and only if its minimum M -eigenvalue is positive; an elasticity Z-tensor is a nonsingular elasticity M -tensor if and only if the minimum M -eigenvalue is positive [5].…”