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

Abstract

M-eigenvalues of fourth-order partially symmetric tensors play important roles in the nonlinear elastic material analysis and the entanglement problem of quantum physics. In this paper, we introduce M-identity tensor and establish two M-eigenvalue inclusion intervals with n parameters for fourth-order partially symmetric tensors, which are sharper than some existing results. Numerical examples are proposed to verify the efficiency of the obtained results. As applications, we provide some checkable sufficient conditions for the positive definiteness and establish bound estimations for the M-spectral radius of fourth-order partially symmetric nonnegative tensors.

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

“…The strong ellipticity condition holds for an elasticity tensor if and only if its minimum M -eigenvalue is positive [17]. Denote σ M (A) as the set of all M -eigenvalues of A. Tensors with special structures, such as nonnegative tensors and M -tensors, are becoming the keynote in recent research [4,5,9,14,21,26,27,28,29]. Particularly, some important properties of M -tensors and nonsingular M -tensors have been established in [4,28].…”

Sharp upper bounds on the maximum $M$-eigenvalue of fourth-order partially symmetric nonnegative tensors

“…where [m] = {1, 2, • • • , m} and [n] = {1, 2, • • • , n}. Such this fourth-order partially symmetric tensor is useful in nonlinear elastic material analysis [1,2,3,5,8,12,15,18,24] and entanglement problem of quantum physics [4,7]. For example, a fourth-order partially symmetric nonnegative tensor with n = 2 or 3, can be used in the two/three-dimensional field equations for a homogeneous compressible nonlinearly elastic material for static problems without body forces [13].…”

“…Preliminaries. In this section, we firstly introduce some definitions and important properties of fourth-order partially symmetric tensors [12,18].…”

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

“…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].…”