AIMS Mathematics, 2020, 5(4): 3664-3681. doi: 10.3934/math.2020237

Research article

Export file:

Format

• RIS(for EndNote,Reference Manager,ProCite)
• BibTex
• Text

Content

• Citation Only
• Citation and Abstract

The generalized conjugate direction method for solving quadratic inverse eigenvalue problems over generalized skew Hamiltonian matrices with a submatrix constraint

1 School of Mathematics and Informatics & FJKLMAA, Fujian Normal University, Fuzhou 350117, P. R. China
2 School of Technology, Fuzhou University of International Studies and Trade, Fuzhou 350202, P. R. China

## Abstract    Full Text(HTML)    Figure/Table

In this paper, we consider a class of constrained quadratic inverse eigenvalue Problem 1.1. Then, a generalized conjugate direction method is proposed to obtain the generalized skew Hamiltonian matrix solutions with a submatrix constraint. In addition, by choosing a special kind of initial matrices, it is shown that the unique least Frobenius norm solutions can be obtained consequently. Some numerical results are reported to demonstrate the efficiency of our algorithm.
Figure/Table
Supplementary
Article Metrics

# References

1. Y. Yang, J. Han, H. Bi, et al. Mixed methods for the elastic transmission eigenvalue problem, Appl. Math. Comput., 374 (2020), 125081.

2. J. Han, Nonconforming elements of class L2 for Helmholtz transmission eigenvalue problems, Discrete Cont. Dyn-B., 23 (2018), 3195-3212.

3. H. Dai, Z. Z. Bai, Y. Wei, On the solvability condition and numerical algorithm for the parameterized generalized inverse eigenvalue problem, Siam J. Matrix Anal. A., 36 (2015), 707-726.

4. K. Ghanbari, A survey on inverse and generalized inverse eigenvalue problems of jacobi matrices, Appl. Math. Comput., 195 (2008), 355-363.

5. Y. X. Yuan, H. Dai, A generalized inverse eigenvalue problem in structural dynamic model updating, J. Comput. Appl. Math., 226 (2009), 42-49.

6. K. W. E. Chu, M. Li, Designing the Hopfield neural network via pole assignment, Int. J. Syst. Sci., 25 (1994), 669-681.

7. Z. J. Bai, The inverse eigenproblem of centrosymmetric matrices with a submatrix constraint and its approximation, Siam J. Matrix Anal. A., 26 (2005), 1100-1114.

8. Z. J. Bai, The solvability conditions for the inverse eigenvalue problem of Hermitian and generalized skew-Hamiltonian matrices and its approximation, Inverse Probl., 19 (2003), 1185-1194.

9. L. F. Dai, M. L. Liang, Generalized inverse eigenvalue problem for (P, Q)-conjugate matrices and the associated approximation problem, Wuhan Univ. J. Nat. Sci., 21 (2016), 93-98.

10. Y. Q. Gao, P. Wei, Z. Z. Zhang, et al. Generalized inverse eigenvalue problem for reflexive and antireflexive matrices, Numer. Math. J. Chin. Univ., 34 (2012), 214-222.

11. P. Wei, Z. Z. Zhang, D. X. Xie, Generalized inverse eigenvalue problem for Hermitian generalized Hamiltonian matrices, Chin. J. Eng. Math., 27 (2010), 820-826.

12. R. H. Mo, W. Li, The inverse eigenvalue problem of hermitian and generalized skew-Hamiltonian matrices with a submatrix constraint and its approximation, Acta Math. Sci., 31 (2011), 691-701.

13. J. Cai, J. Chen, Iterative solutions of generalized inverse eigenvalue problem for partially bisymmetric matrices, Linear Multilinear A., 65 (2017), 1643-1654.

14. J. Cai, J. Chen, Least-squares solutions of generalized inverse eigenvalue problem over HermitianHamiltonian matrices with a submatrix constraint, Comput. Appl. Math., 37 (2018), 593-603.

15. H. C. Chen, Generalized reflexive matrices: Special properties and applications, Siam J. Matrix Anal. A., 19 (1997), 140-153.

16. J. Qian, R. C. E. Tan, On some inverse eigenvalue problems for Hermitian and generalized Hamiltonian/skew-Hamiltonian matrices, J. Comput. Appl. Math., 10 (2013), 28-38.

17. D. Xie, N. Huang, Q. Zhang, An inverse eigenvalue problem and a matrix approximation problem for symmetric skew-hamiltonian matrices, Numer. Algorithms, 46 (2007), 23-34.

18. R. H. Mo, W. Li, An inverse eigenvalue problem for Hermitian and generalized skew-Hamiltonian matrices with a submatrix constraint and its approximation, Acta Math. Sci., 31 (2011), 691-701.

19. W. R. Xu, G. L. Chen, X. P. Sheng, Analytical best approximate Hermitian and generalized skewHamiltonian solution of matrix equation AXAH + CYCH = F, Comput. Math. Appl., 75 (2018), 3702-3718.

20. Y. Liu, Y. Tian, Y. Takane, Ranks of Hermitian and skew-Hermitian solutions to the matrix equation AXA=B, Linear Algebra Appl., 431 (2009), 2359-2372.

21. M. Wei, Q. Wang, On rank-constrained Hermitian nonnegative-definite least squares solutions to the matrix equation AXAH = B, Int. J. Comput. Math., 84 (2007), 945-952.