The Mixed Structural Solutions Of Generalized Sylvester Matrix And Tensor Equations

Quaternions and generalized Sylvester equations play important role in quantum me-chanics,computer science,vector sensor signal processing,aerospace control,pattern recog-nition and other fields.Tensor equations can be used to simulate material behavior in engi-neering sciences,and can also be used to study continuum mechanics.According to the char-acteristics of some structural matrices,the mixed structural solutions of generalized Sylvester matrix equations and tensor equations are studied by using the decomposition technique of matrices or tensors.This thesis is mainly composed of five chapters,with specific contents as follows:In chapter 1,we mainly introduce the research background and current situation of matrix equations and tensor equations,put forward the main research contents of this article,and give some basic concepts and lemmas.In chapter 2,for a class of complex systemΣ_i=1^pA_iX B_i+Σ_j=1^qC_jY D_j=E with unknown matrices X and Y,the definition of tridiagonal-arrowhead pairs of matrices with same elements are given,and problems of two structure solution of the system and its optimal approximation are discussed.By using the specific structure of a tridiagonal and arrowhead matrix,compact form of their vectorize is constructed,and the equation with constraints can be converted to an unconstrained equation resort to Kronecker product of matrices.Then the necessary and sufficient condition for the existence pairs of structure matrices with same elements（X,Y）of the system and its general solution expression are obtained.Meanwhile under the condition of the solution set is not empty,the expression of the optimal approximation to the given tridi-agonal matrix M and arrowhead matrix N is derived by using matrix blocking and properties of the norm of complex matrix.In chapter 3,a class of M self-conjugate mixed structure solutions of the quaternion gen-eralized Sylvester matrix equation AX-Y B=C is discussed,where X is a unitary similar block diagonal M self-conjugate matrix and Y is a self-conjugate matrix.According to the characteristics of the proposed structure matrices,the original structural equation is trans-formed into an equivalent system of unconstrained set of equations.The sufficient conditions of the equation system and the theoretical expression of its general solution are obtained.Thus,the M self-conjugate mixed structure solutions of the original equation are obtained.In particular,the expression for the matrix equation AX=C to have unitary similar block diagonal M self-conjugate solution is derived.When M=0,the constrained mixed structure solution sets of generalized Sylvester equation satisfying||Y||=min are obtained by using the CCD-Q decomposition of quaternion matrix pairs.In chapter 4,the mixed structural solutions of the quaternion Sylvester tensor equation（?）*_NX-Y*_NB=C in the sense of Einstein product are discussed,where Xand Yare un-known Hermitian tensor and Skew-Hermitian tensor respectively.By using the property of the conjugate transpose of quaternion tensor,the original structural equation can be transformed into an equivalent unconstrained tensor equation group.According to the M-P generalized in-verse of the quaternion tensor,the sufficient conditions and general solution expressions of the equivalent equations are obtained,so as to obtain the mixed structure solutions of the original equation.In particular,the conditions and expressions of the tensor equations（?）*_NX=C andY*_NB=-C with Hermitian and Skew-Hermitian tensor solutions are derived.In chapter 5,we summarize the main research work of this thesis and propose some follow-up research directions.