Least-squares solutions of generalized inverse eigenvalue problem over Hermitian-Hamiltonian matrices with a submatrix constraint

In this paper, a gradient-based iterative algorithm is proposed for finding the least-squares solutions of the following constrained generalized inverse eigenvalue problem: given X is an element of C-nxm, Lambda = diag(lambda(1), lambda(2),...,lambda(m)) is an element of C-mxm, find A*, B* is an element of C-nxn, such that parallel to AX -BX Lambda parallel to is minimized, where A*, B* are Hermitian-Hamiltonian except for a special submatrix. For any initial constrained matrices, a solution pair (A*, B*) can be obtained in finite iteration steps by this iterative algorithm in the absence of roundoff errors. The leastnorm solution can be obtained by choosing a special kind of initial matrix pencil. In addition, the unique optimal approximation solution to a given matrix pencil in the solution set of the above problem can also be obtained. A numerical example is given to show the efficiency of the proposed algorithm.