Least-squares solutions of generalized inverse eigenvalue problem over Hermitian–Hamiltonian matrices with a submatrix constraint - Cai, Jing; Chen, Jianlong (Computational & applied mathematics, 2018)
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 ∈ C n × m , Λ = diag ( λ 1 , λ 2 , … , λ m ) ∈ C m × m , find A ∗ , B ∗ ∈ C n × n , such that ‖ A X - B X Λ ‖ 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.