The Least Squares Solution Of The Inverse Problem Of Generalized Anti Centrosymmetric Matrices And Its Generalization

In the research of computational mathematics in recent years,the discussion and specialized analysis of the inverse problems of matrix equations are increasing,which are widely used in statics,big data processing,machine learning,system identification,vibration theory and other fields.In this project,the least squares solution of the generalized anticentric symmetric matrix inverse problem and its generalization problem will be systematically studied.In previous studies on the inverse problem of matrix equations,little attention has been paid to generalized antisymmetric matrices,and the discussion of generalized forms A₁X₁+A₂X₂+…+A_lX_l=Z generally lacks generalized proof details.Since the form of the coefficient matrix of low dimensions and the high dimensional constant are basically the same,this paper starts from the study of binary matrix equations,specifically in the matrix equation AX+BY=Zby giving X,Y,Z and constraining the matrix properties of A,Bto find matrices A,BWhen the equation AX+BY=Z problem has sufficient necessary conditions for solving and the set of solutions that satisfy the matrix equation,on this basis,the best approximate solution in the solution set is solved.Through the research process and core lemma of AX+BY=Z,the solution set and best approximation solution of A₁X₁+A₂X₂+…+A_lX_l=Z are completed.In the study,it is first necessary to require the solution A and B under the generalized anticenter symmetry constraint of the matrix equation AX+BY=Z inverse problem.The method used is to use the properties of generalized anticentric symmetric matrices to decompose the matrix to obtain the antisymmetric transformation matrix,simplify the Primitive matrix equation by the spectral decomposition technology of the antisymmetric transformation matrix,and find the matrices A and B when the equations AX+BY=Z problem have sufficient necessary conditions for solving and the set of solutions satisfying the matrix equations.Then,on this basis,the calculation method of Frobenius norm and the least squares solution of the matrix are used to find the best approximate solution and give a supporting numerical example.Based on the results of the comprehensive analysis of the solution under the generalized anticentric symmetry constraint of the AX+BY=Z inverse problem,the solution under the generalized anticentric symmetry constraint of the matrix equation A₁X₁+A₂X₂+…+A_lX_l=Z inverse problem is studied Ai（i=1,...l）.The method used is to use the properties of the generalized anticentric symmetric matrix to decompose the matrix to obtain the antisymmetric transformation matrix,simplify the original matrix equation by the spectral decomposition technology of the antisymmetric transformation matrix,and find the matrix Ai（i=1,...l）When the equation A₁X₁+A₂X₂+…+A_lX_l=Z has sufficient necessary conditions for solving the problem and a set of solutions satisfying the matrix equation.Then,on this basis,the calculation method of Frobenius norm and the least squares solution of the matrix are used to find the best approximate solution and give a supporting numerical example.AX+BY=Z matrix equation and its promotion form are a kind of mathematical problems closely related to practical applications.Especially in the post information era with rich big data reserves,the rapid development of large models of natural processing in artificial intelligence has led to a more specific demand for matrix and equation operations in computational mathematics,Therefore,the proposal and demonstration of the analytical and approximate solutions of the AX+BY=Z matrix equation and its extended form have more important practical application value.