The Solutions Of Several Kinds Of Special Matrix Equations And The Procrustes Problems

Inverse problems of matrices and constrained matrix equations have been an active and important research topic in the field of computational mathematics and applied mathematics.They have been of interest for many applications,including structural dynamics,solid mechanics,nanomaterial mechanics,autocontrol system,stochastic control and vibration control.Some inverse eigenvalue problems and matrix equations problems under different matrices have been systematically studied in this M.S.thesis.The main work and research results are as follows:1.According to the system of linear algebraic equations Ax = b,the ma-trix decomposition is used to construct a kind of special five and seven diagonal matrix decomposition of symmetric positive definite matrices.This paper gives a sufficient and necessary condition for the solvability of inver-se problems of the matrices and the general representation.The algorithms and numerical examples are given.2.An efficient method based on the properties of circulant matrices a-nd the optimization theory is used to solve the least squares problem of high-dimensional linear matrix equations ArXBr = Cr(r=1,2,...,m)over the constraint of Hankel-circular matrices and Hankel-anti-circular matrices X.The general expression of the solution is given by the linear equations derived from the matrix equations.WhenAl,A2,L.Ap.Bp+1,L,Bm are permuta-tion matrices,it is proved that the coefficient matrix of linear equations is a special symmetric circulant matrix.The determination condition of the unique solution and its expression are derived,The relevant condition with unique solutions is obtained by the m1-norm of the matrices.Finally,resul-ts of numerical experiments are given.3.The symmetric solutions of the nonlinear matrix equation AX2 =B ar-e discussed.The sufficient and necessary condition of this problem and its partial solutions are given by the properties of symmetric positive semi-definite matrices and the previous research achivements.Furthermore,the expression of the whole solutions is studied.The algorithms and nume-rical results are presented.