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 $X\in C^{n\times m}$, $\Lambda =diag({\lambda }_1,{\lambda }_2,\dots ,{\lambda }_m)\in C^{m\times m}$, find $A^{\ast },B^{\ast }\in C^{n\times n}$, such that $\parallel AX-BX\Lambda \parallel$ is minimized, where $A^{\ast },B^{\ast }$ are Hermitian–Hamiltonian except for a special submatrix. For any initial constrained matrices, a solution pair $(A^{\ast },B^{\ast })$ can be obtained in finite iteration steps by this iterative algorithm in the absence of roundoff errors. The least-norm 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.