## Abstract

In this paper, the inverse eigenvalue problems for skew-Hermitian reflexive and anti-reflexive matrices and their associated optimal approximation problems which are constrained by their partially prescribed eigenpairs are considered, respectively. First, the necessary and sufficient conditions of the solvability for the inverse eigenvalue problems of skew-Hermitian reflexive and anti-reflexive matrices are both derived, and the general solutions are also presented. Then the solutions of the corresponding optimal approximation problems in the Frobenius norm to a given matrix are also given, respectively. Furthermore, we give the algorithms to compute the optimal approximate skew-Hermitian reflexive and anti-reflexive solutions and present some illustrative numerical examples.

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## Acknowledgements

The first author is supported by joint research project of Laurent Mathematics Center of Sichuan Normal University and National-Local Joint Engineering Laboratory of System Credibility Automatic Verification. The second author is supported by Science and Technology Commission of Shanghai Municipality (Grant No. 18dz2271000).

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Communicated by Jinyun Yuan.

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Xu, W., Chen, G. Inverse eigenvalue problems for skew-Hermitian reflexive and anti-reflexive matrices and their optimal approximations.
*Comp. Appl. Math.* **39, **184 (2020). https://doi.org/10.1007/s40314-020-01208-5

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### Keywords

- Inverse eigenvalue problem
- Optimal approximation problem
- Skew-Hermitian reflexive matrix
- Skew-Hermitian anti-reflexive matrix

### Mathematics subject classification

- 65F18
- 15A51
- 15A18
- 15A12