Linear Constrained Least Squares Solutions Of Matrix Equation CZC～T=T

This paper deals with the problem of finding a solution of a constrained least squares problem from the solution set of its unconstrained problem. We focus on the least squares problem of matrix equation CZ C^t = T and its some linear constrained problems including min_X,Y ||AXA^T + BYB^T - T‖_F and its symmetric or skew-symmetric solutions. By transforming the right-hand side matrix T, we show that solutions of these considered constrained problems can be found from the solution set of the unconstrained least squares problem with a modified right-hand side matrix. This approach leads to simple representations of general solutions to the constrained problems in form, then least norm solutions follow from the simple formulae proposed easily. The paper consists of eight chapters.In the first chapter, we give some introductions of least squares problems of linear matrix equation.In the second chapter, we review the simple representations of the solutions to matrix least squares problem min_Z ‖CZC^T — T‖_F As a simple example, we show that how to find the general representations of solutions to symmetric or skew-symmetric problems from the unconstrained solution set.In the third chapter, we consider the necessary and sufficient conditions for the existence of a block diagonal solution Z = diag(X, Y) to least squares problem minz ‖CZC^t‖_f. When these conditions are satisfied, the representations of the block diagonal solutions X and Y are given.In the fourth chapter, we illustrate how to transform the right-hand side matrix T equivalently that leads to those conditions are released, then the general solutions and least norm solutions of the unconstrained problem minx,y || AXA^T + BYB^T — T‖_F are simply formulated respectively.In the fifth chapter, we formulate the symmetric and skew-symmetric solutions of least squares problem uan_X,Y || AO^T + BYB^T - T‖_F based on the general solutions of its unconstrained problem.In the sixth chapter, we give the symmetric and skew-symmetric least norm solutions of matrix least squares problem min_X,Y U-AXA^T + BYB^T - T‖_F directly.