Research On The Z-Eigenvalue Inclusion And Bounds On Z-Spectral Radii Of Tensors

With the rise of quantum computing,machine learning,artificial intelligence and other fields,tensors have been applied to solve many problems,tensors have attracted many scholars` attention.In 2005,Prof.Lek-heng Lim and Prof.Qi Liqun defined the eigenvalues of tensors respectively.In recent years,there are many eigenvalue problems applied in practical application,such as automatic control,hyperspectral theory,nuclear magnetic resonance imaging,high-order markov chain,optimal rank one approximation in data analysis,etc.In particular,the Z-spectral radius of the tensor plays an important role in high-order markov chain and optimal rank one approximation of quantum entanglement.Solving eigenvalue problems of tensors is the same as solving higher order equations.Therefore,the computation of eigenvalue will be very difficult.It is very essential to estimate the range of eigenvalues.Scholars focused on the estimation of eigenvalues bounds of higher-order tensors.In this paper,the bounds of Z-eigenvalues is mainly concerned.The work is as follows:1.The Z-eigenvalue inclusion sets of tensors are given.The results include general tensors and Z-eigenvalue inclusions of weakly symmetric nonnegative irreducible tensors.Finally,based on the result of eigenvalue inclusion sets,the bounds for Z-spectral radius of tensors are given.2.Using the elements of tensor A and B,calculating the Hadamard product bounds for the Z-eigenvalue spectral radius of tensors.