Sharp Bounds on the Minimum M-Eigenvalue of Elasticity M-Tensors

Abstract

The M-eigenvalue of elasticity M-tensors play important roles in nonlinear elastic material analysis. In this paper, we establish an upper bound and two sharp lower bounds for the minimum M-eigenvalue of elasticity M-tensors without irreducible conditions, which improve some existing results. Numerical examples are proposed to verify the efficiency of the obtained results.where (A · xy 2 ) i = ∑ j,k,l∈ [n] a ijkl x j y k y l , (Ax 2 y·) l = ∑ i,j,k∈ [n] a ijkl x i x j y k , then the scalar λ is called an M-eigenvalue of the tensor A, and x and y are called left and right M-eigenvectors of A associated with the M-eigenvalue. Then the M-spectral radius of A is denoted byRecently, tensors with special structures, such as nonnegative tensors, M-tensors and H-tensors, are becoming the focus in recent research [2,24,[28][29][30][31]. Some effective algorithms for finding eigenvalue and the corresponding eigenvector have been implemented [1,24,[32][33][34][35]. For example, Bozorgmanesh et al. [32] propose an algorithm that can solve E-eigenvalue problem faster. However, it is very difficult for these algorithms to compute all M-eigenvalues or E-eigenvalues. Thus, some researchers turned to investigating eigenvalue inclusion sets [4,7,[36][37][38][39][40][41]. Particularly, some bounds for the minimum H-eigenvalue of nonsingular M-tensors have been proposed [2,28,30,42,43]. Ding et al. [24] introduced a structured partially symmetric tensor named elasticity M-tensors and established important properties of elasticity M-tensors and nonsingular elasticity M-tensors.