An Abstract Iterative Algorithm For Solving The Matrix Equation And Its Least Squares Problem With Applications

The problem of solving the matrix equation with constraints is an important sub-ject in the context of numerical algebra,which is intended to find the solution of amatrix equation in a matrix set with certain constrains. The problem may vary greatlywith different constrains and different matrix equations. For example, given matrixA,B,X0, a typical problem of matrix equation with constrains, is to find with cer-tain constrains ,such that; if the matrix equation AX = B is incon-sistent, we can recast the original problem as finding with certain constraints suchthat , which is also called the least squares problem. One can alsoconsider the optimal approximation problem, for the problem.The matrix equation problems are widely occurred in biology, electricity, molec-ular spectroscopy, vibration theory, finite elements, structural design, solid mechanics,parameter identification, automatic control theory, linear optimal control and so on.And because of the lots of different problems in these areas, the theory of matrix equa-tion with constraints developed quickly, the problem of solving the matrix equationwith constraints is a hot spot of the computational mathematics.The methods of solving the matrix equation with constraints are matrix factor-ization algorithm and iterative algorithm, it needs to construct different formulas andalgorithms for different equations. This thesis focuses on studying the matrix equa-tion problem systematically, and proposed an abstract algorithm of solving the matrixequation with constraints, and established a strict convergence theory. Using this algo-rithm, we can solve the sets of matrix equation satisfying some constraint conditions,such as symmetric, antisymmetric, centrosymmetric, centroskew symmetric, re?exive,antire?exive, bisymmetric, symmetric and antipersymmetric, symmetric orthogonalsymmetric, symmetric orthogonal antisymmetric, Hermite generalized Hamilton ma-trix;So we can solve the problem with this algorithm, if the set of constrain matrixcan make a subspace in matrix space, and this algorithm also can solve the optimalapproximation and least squares problem. So this abstract algorithm has universal andimportant practical value.