Several Constrained Solutions Of Matrix Equation AX=B And AXB=C

The constrained matrix equations are problems to find the solutions of matrix equations in contrained matrix sets. It is one of the most important problem in numerical algebra. And is widely used in structural design, system identification, automatics control, vibration theory. In this M.S. thesis, the following problems are considered mainly.1. Using the generalized inverse, singular value decomposition, tensor product and draw operator of matrices, the sufficient and necessary conditions for the existenceof and the general expressions for the matrix equation AX = B has a rotation invariable solution or a circulant solution are derived. In the solution set of the matrix equation, the unique optimal approximation solution to a given matrix in Frobenius norm is given. Numerical algoritms and numerical experiments to comput the unique optimal approximation solution are also presented.2. Let P∈R^n×n and Q∈R^m×n are selfadjoint involutory matrices (i.e. P^H = P, P² = I_n, Q^H = Q, Q² = I_m). An n×m complex A is said to be a generalized centro (or centroskew) symmetric matrix with respect to the selfadjoint involutory matrices P and Q if A = PAQ (or A = -PAQ). By using the generalized singular-value decompositions of the matrix pair, this paper has been established the necessary and sufficient conditions for the existence of and the expressions for the generalized centro (or centroskew) symmetric solution of the matrix equation AXB = C, showed that there is an unique optimal approximation solution to a given matrix in Frobenius norm in the generalized centro (or centroskew) symmetric solution set of matrix equation AXB = C, and presented numerical algoritms and numerical experiments to comput the unique optimal approximation solution. In this paper, in addition, has also been given the matrix form LSQR iterative methods to comput a generalized centro (or centroskew) symmetric solution and the unique optimal approximation solution to a given matrix in Frobenius norm of the matrix equation AXB = C.