Constraint Matrix Least Squares Problem Of Equation (s)

The constrained linear matrix equations and related least squares problems have been of interest for many applications in scientific fields, including particle physics and geology, inverse problems of vibration theory, digital image and signal processing photogrammetry, multidimensional approximation.In this paper, we study least squares problems of constrained matrix equations from recursive algorithms and direct methods. The main work is as follows.By applying the generalized singular value decomposition , the expression of the weighted least-square solutions of the matrix equation A^T XB -B^T X^TA = D is obtained. In addition, the expression of the optimal approximation solution to the given matrix is obtained. At the time, by applying the canonical correlation decomposition of matrix pairs, we discuss the symmetric ortho-anti-symmetric solutions of the matrix equation B^T XB = Don some linear manifolds and its optimal approximation solutionSeveral recursive algorithms are presented to solve constrained linear matrix equations. First, we discuss the least squares symmetric solutions of the matrix equations A₁ XB₁ =D₁, A₂XB₂ =D₂. By this recursive algorithm, the solution with leastnorm can be obtained by choosing a special kind of initial symmetric matrix. In addition, the unique optimal approximation solution to a given matrix in Frobenius norm can be obtained by finding the least norm solution of the new minimum residual problem. Second, we discuss the least squares skew-symmetric solutions of matrixequations A^T XA = C,B^T XB = D and its optimal approximation solution. At last, therecursive algorithm to the matrix equation AXB + CXD = F over centrosymmetric solution is also discussed.