Least Square Solutions And Optimal Approximation Solution Of Several Matric Equations On The Linear Manifold

The firstly inverse problem of matrix was raised by the J.B.Keller. There are a series of results about them. By applying singular value decomposition, this paper mainly discuss the following least-square solutions and optimal approximation solution of matrix equations on the linear manifold which are widely used in structure design, building engineering and vibration engineering: ProblemⅠGiven X , B∈R n×n. Find A∈Ssuch that B T AB ? XW= min. ProblemⅡGiven X , B∈R n×m. Find A∈Ssuch that AX ? B= min. ProblemⅡGiven X , B∈C n×k. Find A∈Ssuch that AX ? B= min. ProblemⅣGiven A?∈R n×n (C n×n). Find A*∈SEsuch that * ? min?A ? A = A∈SEA ? A. Where S denotes some linear manifolds, S Edenotes the solution sets of problemⅠor problemⅡor problemⅢ, ? is Frobenius norm, ? W is weighted norm.The main results of this paper are listed as follows:1 The expressions of the weighted least square symmetric orthogonal anti- symmetric solutions of problemⅠon the linear manifold are derived. In addition, the optimal approximation solution is also obtained.2 The expressions for the least-square solutions, which are symmetric orthogonal anti-symmetric matrices and anti-symmetric orthogonal anti-symmetric matrices of problemⅡon the linear manifold are established. In addition, we have presented the optimal approximation solution to a given matrix.3 The expressions for the least-square solutions, which are complex anti-symmetric solutions of problemⅢon the linear manifold are established. In addition, we have presented the optimal approximation solution to a given matrix.