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In this paper, we establish sharp upper and lower bounds on the minimum  M -eigenvalue via the extreme eigenvalue of the symmetric matrices extracted from elasticity  Z -tensors without irreducible conditions. Based on the lower bound estimations for the minimum  M -eigenvalue, we provide some checkable sufficient or necessary conditions for the strong ellipticity condition. Numerical examples are given to demonstrate the proposed results.

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