Further Study on Z-Eigenvalue Localization Set and Positive Definiteness of Fourth-Order Tensors

Abstract

Fourth-order tensors play a fundamental role in signal processing, wireless communication systems, image processing, data analysis and higher-order statistics. In this paper, we introduce a Z-identity tensor and establish two Z-eigenvalue inclusion sets for fourth-order 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 of fourth-order symmetric tensors. Further, we propose upper bounds on the Z-spectral radius of fourth-order nonnegative tensors and estimate the convergence rate of the greedy rank-one algorithms under suitable conditions.