Applications Of Matrix Semi-tensor Product On Algebraic Structures

Matrix semi-tensor product is a new matrix multiplication,which not only breaks the limitation of matrix dimension of traditional matrix multiplication,but also main-tains the main properties of original matrix multiplication.At the same time,it also has the properties of pseudo-commutativity and so on,which are better than before.There-fore,it is a convenient and powerful new mathematical tool.In this paper,some im-portant matrix equation solving problems are studied by using the matrix semi-tensor product method.Some properties of bundled Lie algebras which constructed based on the semi-tensor product theory are also discussed.The thesis includes the following contents:Chapter 1 introduces the research background,significance and present status of semi-tensor product.Chapter 2 introduces some necessary preliminaries on semi-tensor product.Chapter 3 investigates the least squares solutions of matrix semi-tensor product equation AXB=C.Firstly,according to the definition of the semi-tensor product,transform the matrix semi-tensor product equation into an ordinary matrix equation,and then studies the least squares solutions of matrix-vector equation and matrix equa-tion respectively by the derivation of matrix differential operation.Furthermore,the specific form of the least squares solutions is given.Chapter 4 investigates the solvability of matrix semi-tensor product equation AX2=B.Discusses the compatible conditions on matrices A and B when the matrix semi-tensor product equation has solution,and studies the sufficient and necessary condi-tions of solubility.Finally,the specific methods of solving this equation is concreted.Chapter 5 investigates the solvability of matrix second semi-tensor product equa-tion A(?)lX=B.Studies the compatible conditions of matrices A and B and the neces-sary and sufficient condition for the solvability of matrix second semi-tensor product equation successively.Furthermore,the specific methods of solving this equation is provided.Chapter 6 investigates bundled Lie algebra.Discusses the representation and basis of it and some important subalgebras,and explores its properties as a direct limit.Chapter 7 summarizes the research results of this paper,and makes a prospect for the future research work.