Research On Kronecker Product Decomposition Of Matrices And Inverse Eigenvalue Problems Of Tensors Over The Quaternion Field

Quaternion matrices and tensors have extensive practical applications in the fields of physics,face recognition,image processing,and medical diagnosis.In recent years,some decomposition of quaternion matrices and inverse eigenvalue problems of tensors have be-come hot topics in the matrix computation.This thesis mainly uses theoretical tools such as the Kronecker product of quaternion matrices,the properties of quaternion structural tensors,the transformation operators of quaternion tensors,and their Einstein products to study the Kronecker product decomposition and best approximation problems of quaternion matrices,and discuss the inverse eigenvalue problems of Hermitian tensors,Skew-Hermitian tensors and self-conjugate symplectic tensors over the quaternion field.The content of this article is divided into five chapters,and the specific content is as follows:In chapter 1,the research background and development status of the matrix Kroneck-er product decomposition and tensor inverse eigenvalue problems at home and abroad are introduced,and relevant basic concepts,lemmas and other preparatory knowledge are given.In chapter 2,the Kronecker product decomposition of the quaternion matrix A is dis-cussed,and the existence condition and calculation method of the quadratic root of A in the Kronecker product sense are given.When such decomposition does not exist,the correspond-ing best approximation decomposition method and calculation formula are given.In chapter 3,the inverse eigenvalue problem and the best approximation solution of quaternion Hermitian tensors under the Einstein product is discussed.Specifically,given s quaternion tensor characteristic pairs,find the quaternion Hermitian tensor?to include the given characteristic pairs;At the same time,for the given quaternion tensor,the best approximate solution in the solution set of the above inverse problem in the sense of Frobenius norm is obtained.In addition,the Skew-Hermitian solution of a class of quaternion tensor equation and its right eigenvalue inverse problem solution are given.Finally,a numerical example is used to verify the feasibility of the proposed method.In chapter 4,the inverse eigenvalue problem of quaternion self-conjugate symplectic tensors under the Einstein product is discussed.Firstly,the definition of quaternion conjugate symplectic tensors is given using the transformation operator of tensors,and its properties and characteristic structures are discussed.Secondly,for the given finite number of quaternion tensor characteristic pairs,find the quaternion self-conjugate symplectic tensorto include all the given characteristic pairs.As an application,the necessary and sufficient conditions for the existence of the conjugate symplectic tensor solution to the quaternion tensor equation?*_N=and its solution expression are given.Finally,a numerical example is used to verify the feasibility of the proposed method.In chapter 5,we summarize the main current research work and point out the relevant issues that deserve in-depth discussion.