A Number Of Properties Of Singular Value And Higher Order Singular Value Decomposition Of Tensor

Singular value and higher-order singular value decomposition of tensors is widely used in varieties of disciplines.Recently,this theory has drawn attention of many scholars and been popular topic studied by the domestic and foreign experts and scholars.In this thesis,we mainly study the properties of singular value and higher-order singular value decomposition of tensors.Firstly,we extend conclusions that the singular value of matrix is invariat under orthogonal transformation and the positive definiteness of real quadratic form is preserved by contragradient transformation to tensors.Secondly,we study the properties of the singular value of a rectangular tensor in Kronecker product.We define Kronecker summation of tensors and study the properties of its H-eigenvalue.Then we study the properties of the higher-order singular value decomposition of tensors in Kronecker product.Finally,we research the properties of higher-order singular value decomposition of structured tensors.We prove the necessary and sufficient condition that the core tensor of a 3-order 2-dimensional Hankel tensor,which is obtained by higher-order singular value decomposition,is a diagonal tensor.Then we explore that a number of cyclic tensor have confirmed structure of the higher order singular value decomposition.