Some Properties Of Essentially Nonnegative Tensors And Copositive Tensors

Tensors have important applications in many scientific fields, such as signal processing, data analysis and mining and so on. Tensor eigenvalue theory is an important aspect of tensor research. In this thesis, some related properties of essentially nonnegative tensor and copositive tensor are researched.Firstly, a theorem on H++-eigenvalues of symmetric nonnegative tensors is proved by applying the Perron-Frobenius theory of nonnegative tensors. Then, the spectral properties of symmetric nonnegative matrices are generalized to symmetric essentially nonnegative tensors, a maximum property for the largest H-eigenvalue of symmetric essentially nonnegative tensors is proved, some bounds of the eigenvalue are given and a theorem on H++-eigenvalues of symmetric essentially nonnegative tensors is given.Finally, copositive tensors and eigenvalues of copositive tensors are discussed. Criterions for copositive property of symmetric Z-matrices are extended to symmetric Z-tensors, a sufficient and necessary condition is given to verify (strictly) copositive tensors for a symmetric Z-tensor and some further properties of (strictly) copositive tensors are discussed.