An iterative method for obtaining the Least squares solutions of quadratic inverse eigenvalue problems over generalized Hamiltonian matrix with submatrix constraints

In this paper, we consider a class of constrained matrix quadratic inverse eigenvalue problem and its optimal approximation problem. It is proved that the proposed algorithm always converge to the generalized Hamiltonian solutions with a submatrix constraint of Problem 1.1 within finite iterative steps in the absence of roundoff error. In addition, by choosing a special kind of initial matrices, it is shown that the minimum norm solution of Problem 1.1 can be obtained consequently. At last, for a given matrix group in the solution set of Problem 1.1, it is proved that the unique optimal approximation solution of Problem 1.2 can be also obtained. Some numerical results are reported to demonstrate the efficiency of our algorithm.