Bound estimations of bi-block <inline-formula><tex-math id="M1">$ M $</tex-math></inline-formula>-eigenvalues for bi-block symmetric tensors

Abstract

Bi-block symmetric tensors are extensions of elasticity tensors, whose positive definiteness plays an important role in the elasticity theory. In this paper, we establish bi-block $ M $-eigenvalue inclusion intervals of the bi-block symmetric tensors, which are sharper than some existing results. Based on the new sharp inclusion intervals, we give new definitions of bi-block (strictly) diagonally dominant tensor and bi-block symmetric $ B_0 $($ B $)-tensor, and show that both of them are bi-block positive (semi)definite tensors. Then we give some easily checkable sufficient conditions for judging bi-block positive (semi)definiteness of the bi-block symmetric tensor. At last, we give four numerical examples, whose bi-block positive (semi)definiteness can be determined by the new definitions but the existing ones.