Z-eigenvalues based structured tensors: $M_z$-tensors and strong $M_z$-tensors

The positive (semi-)definiteness of even-order tensors has been widely studied in these years due to its applications in various aspects, such as spectral hypergraph theory, automatic control, polynomial theory, stochastic process, magnetic resonance imaging and so on. It has been shown that $M$-tensors, $B$-tensors, $H$-tensors, Hilbert tensors and stochastic tensors can be positive definite under proper conditions. However, there are still many positive definite tensors that can not be determined by the above criteria. In this paper, we provide a new class of positive definite tensors whose non-diagonal entries can be positive compared to (strong) $M$-tensors, and we call it strong $M_z$-tensors, which can arise from discretizing differential equations, since it is based on Z-eigenvalues rather than H-eigenvalues traditionally. Moreover, we show that an even-order (strong) $M$-tensor must be an (a strong) $M_z$-tensor, which reflects the inclusion relationship between even-order $M$-tensors and $M_z$-tensors. We also introduce (strong) $H_z$-tensors, as a generalization of (strong) $M_z$-tensors, and its positive semi-definiteness (positive definiteness) has been studied. Finally, some conditions are given for a tensor to be an (a strong) $M_z$-tensor and we use it to study the stability of a high-order nonlinear system.