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

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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.

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Acknowledgments

This research was supported by the National Natural Science Foundations of China (No. 61473332) and the Natural Science Foundation of Anhui Province (No. 1508085MA12). The authors are grateful to the editor and the anonymous referees for their valuable comments, which help to improve this paper.

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Authors and Affiliations

  1. School of Science, Huzhou University, Huzhou, 313000, Zhejiang, People’s Republic of China

    Jing Cai

  2. Department of Mathematics, Southeast University, Nanjing, 211189, Jiangsu, People’s Republic of China

    Jing Cai & Jianlong Chen

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  1. Jing Cai
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  2. Jianlong Chen
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Correspondence to Jing Cai.

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Communicated by Jinyun Yuan.

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Cai, J., Chen, J. Least-squares solutions of generalized inverse eigenvalue problem over Hermitian–Hamiltonian matrices with a submatrix constraint. Comp. Appl. Math. 37, 593–603 (2018). https://doi.org/10.1007/s40314-016-0363-3

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  • DOI: https://doi.org/10.1007/s40314-016-0363-3

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