Brualdi-type New Eigenvalue And Singular Value Inclusion Sets Of Tensors

In some applications,it is not necessary to know all the eigenvalues of the matrix,only need to know what range the eigenvalues are in,so the estimation of the eigenvalues is particularly necessary.There are three famous eigenvalues inclusion sets of matrix in matrix theory: Ger(?)gorin-type disk theorem,Brauer-type theorem,and Brualdi-type theorem,the result of the Brualdi-type theorem is the most accurate.They have been generalized to tensors,we mainly study the Brualdi-type inclusion sets of square tensor eigenvalues and rectangular tensor singular values in this paper.Using two different ways,we present the Brualdi-type eigenvalue inclusion sets characterized by the girth of the digraph associated with a square tensor,and the Brualdi-type eigenvalue inclusion sets of the general square tensor.As applications,using these two new Brualdi-type tensor eigenvalue inclusion sets,a method for determine the nonsingularity of tensors and the sufficient conditions for the positive definiteness of the even order real symmetric tensors are given.Using two different ways,we present the Brualdi-type singular value inclusion sets by using the girth of the sub-digraph of digraph associated with a rectangular tensor,and three singular value inclusion sets of the rectangular tensor obtained by the relation between the singular value of the rectangular tensor and the eigenvalue of the lifting square tensor.Using the digraph associated with a rectangular tensor,two bounds for the largest singular value of nonnegative rectangular tensors are given.