A New Class Of Structured Tensors-SB-tensors

Tensor theory is an attention-getting research subject in the field of the multiple numerical linear algebra,in which structure tensor is one of the main topics.Real symmetric tensors,as a kind of special structure tensor,has been widely used in many fields such as the polynomial theory,automatic control theory,magnetic resonance imaging,spectral hypergraph theory.The identifying of the positive definiteness(positive semi-definiteness)of real symmetric tensor is often involved in these applications.In this thesis,a new kind of structure tensors:SB-tensors,is proposed and some properties of SB-tensors are given.Then the relationships of SB-tensors with several existing structure tensors(for example.B-tensors,DB-tensors and MB-tensors)are discussed.Thus we get new sufficient conditions of positive definite(semi-definite)of real symmetric tensors,which provides a criterions to judge the positive definiteness(semi-definiteness)for the even order real symmetric tensor.Finally,the decomposition of SB-tensors is studied,and a decomposition form of the real symmetric SB-tensor is given,and another proof of the positive definiteness(semi-definiteness)of the even order real symmetric SB-tensor is given by using the decomposition form.