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

Abstract

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 \(X\in C^{n\times m}\), \(\Lambda =\mathrm{diag}(\lambda _1,\lambda _2,\ldots ,\lambda _m)\in C^{m\times m}\), find \(A^*,B^*\in C^{n\times n}\), such that \(\Vert AX-BX\Lambda \Vert \) 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 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.