Finite iterative algorithm for the symmetric periodic least squares solutions of a class of periodic Sylvester matrix equations

The present work proposes a finite iterative algorithm to find the least squares solutions of periodic matrix equations over symmetric -periodic matrices. By this algorithm, for any initial symmetric -periodic matrices, the solution group can be obtained in finite iterative steps in the absence of round-off errors, and the solution group with least Frobenius norm can be obtained by choosing a special kind of initial matrices. Furthermore, in the solution set of the above problem, the unique optimal approximation solution group to a given matrix group in the Frobenius norm can be derived by finding the least Frobenius norm symmetric -periodic solution of a new corresponding minimum Frobenius norm problem. Finally, numerical examples are provided to illustrate the efficiency of the proposed algorithm and testify the conclusions suggested in this paper.