Moore-Penrose Inverse Of Dual Matrices

The dual r-rank decomposition and its applications,the QLY least-squares and the QLY least-squares minimal-norm of linear dual least squares problems,the perturbation of MoorePenorse inverse and dual Moore-Penorse generalized inverse are mainly studied in this thesis.The full thesis is divided into five parts.In the first part,we introduce the generalized inverse theory,the development history and research status of matrix perturbation,dual Moore-Penrose generalized inverse and their applications.In the second part,we introduce some symbols and some basic concepts and lemmas of the generalized inverse theory,matrix perturbation and dual Moore-Penrose inverse.In the third part,we discuss the dual r-rank decomposition of dual matrix,get its existence conditions and equivalent forms of the decomposition.Then we derive some characterizations of the dual Moore-Penrose generalized inverse(DMPGI).Based on DMPGI,we introduce one special dual matrix(dual EP matrix).By applying the dual r-rank decomposition,we derive several characterizations of the dual EP matrix,dual idempotent matrix,dual generalized inverses,and relationships among dual Penrose equations.In the fourth part,we introduce the definition of the QLY total order comparing the magnitude of the dual vector of the same order.Then we consider the QLY least-squares problem and give its compact formula.Furthermore,by applying the QLY total order,we study the QLY least-squares minimal-norm problem and give its compact formula based on the QLY least-squares problem.Meanwhile,by comparing with the analogue of the leastsquares and the least-squares minimal-norm solution,we can always investigate the QLY least-squares and the QLY least-squares minimal-norm of linear dual least sqaures problems.In particular,in the presence of analogue of the least-squares solution,we can get a general QLY least-squares solution with less error under the QLY total order.In the fifth part,we present explicit expressions for the Moore-Penrose inverse of the matrix with perturbation under the rank condition in the real field.Then we calculate the error between the Moore-Penrose inverse and the DMPGI and give an upper bound of this error.Furthermore,we discuss the specific expressions of the Moore-Penrose inverse of the perturbation matrix under some special range conditions,and give the error and upper bound of the Moore-Penrose inverse and the dual Moore-Penrose inverse.