Further Studies Of The Rectangular Tensors

Tensors are generalizations of the matrices and an important tool in physics and continuum mechanics.In recent years,In the study of the strong ellipticity condition problem in solid mechanics and the entanglement problem in quantum physics,the rectangular tensors were introduced.In this paper,we give the definitions of three conditions on rectangular tensors upon the properties of square tensors.Some properties of square tensors are extended to rectangular tensors.The full text is divided into four parts.In the first part,we briefly introduce the research background and current situation of the tensor,and probe into the main content of the paper.In the second part,we recall the basic concepts and properties of tensors.In the third part,we give the definitions of Condition(P)and Condition(PO),and study the properties of the rectangular tensors when they satisfy the two conditions.The main results:Theorem 3.2 Suppose that A=(ai1…ip,j1…jp)is a(p,q)-th order m×n dimensional partially symmetric rectangular tensor,p and q are even.(a)A satisfies Condition(P)if and only if it is positive definite.(b)A satisfies Condition(P0)if and only if it is positive semi-definite.Theorem 3.4 When at least one of p and q is odd,there does not exist a rectan-gular tensor which satisfies Condition(P).Theorem 3.5 Assume that A =(ai1…jp,j1…jq)is a(p,q)-th order m×n dimensional rectangular tensor,then A satisfies Condition(P)if and only if for nonzero vectors x ∈ Rm,y ∈ Rn,there exists two positive diagonal matrices Sx =diag(s1,S2,…,Sm),Ty =diag(t1,t2,…,tn)such that xTSx(Axp-1yp)>0,yTTy(Axpyq-1)>0.In the fourth part,we definite the Condition(R),and extend the properties of strictly nonnegative tensors to the rectangular tensors.Theorem 4.3 Suppose that A is a(p,q)-th order m×n dimensional nonnegative rectangular tensor.If at least one of A(eip,·),A(·,ejq)is irreducible,then A satisfies Condition(R).