NEW Z-EIGENVALUE LOCALIZATION SETS FOR TENSORS WITH APPLICATIONS

In this paper, let A = (a(i1i2) ... i(m)) is an element of R-[m,R-n], when m >= 4, based on the condition parallel to x parallel to(2) = 1, a new Z-eigenvalue localization set for tensors is given. And we extend the Gershgorin-type localization set for Z-eigenvalues of fourth order tensors to higher order tensors. As an application, a sharper upper bound for the Z-spectral radius of nonnegative tensors is obtained. Let H be a k-uniform hypergraph with k >= 4 and A(H) be the adjacency tensor of H, a new upper bound for the Z-spectral radius rho(H) is also presented. Finally, a checkable sufficient condition for the positive definiteness of even-order tensors and asymptotically stability of time-invariant polynomial systems is also given.