Some Properties Of Copositive Tensors Via Einstein Product

With the coming of the big data era,high-order high-dimensional tensors received much attentions of researchers' in recent years,and now they are developed into a new research branch in mathematics named multilinear algebra.In numerical algebra,the tensor is the main research object,and its related problems have become a hot topic in recent years,especially the determination of structured tensors and the eigenvalue problem of high order tensors.As a special kind of structured tensor,the copositive tensor receives a special concern due to its wide applications in vacuum stability of a general scalar potential,polynomial optimization,tensor complementarity problem and tensor eigenvalue complementarity problem.We mainly study a class of important structured tensors-copositive tensor that is not easy to identify.and explore the spectral theories for Einstein product of tensors in the thesis,We first give some properties of the(generalized)inverse of a copositive tensor,and discuss some spectral properties of copositive tensor.Secondly,we introduce the definition of balanced subspace and study its related properties.Finally,we further discuss some properties and necessary and sufficient conditions of copositive tensor under orthogonal projection tensor.