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A System of Coupled Quaternion Matrix Equations with Seven Unknowns and Its Applications

  • Zhuo-Heng HeEmail author
Article
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Abstract

In this paper, a system of coupled quaternion matrix equations with seven unknowns
$$\begin{aligned} A_{i}X_{i}+Y_{i}B_{i}+C_{i}Z_{i}D_{i}+F_{i}WG_{i}=E_{i} \end{aligned}$$
is considered, where \(A_{i},B_{i},C_{i},D_{i},F_{i},G_{i}\) and \(E_{i}\) are given matrices, \(X_{i},Y_{i},Z_{i}\) and W are unknowns \((i=1,2)\). Some practical necessary and sufficient conditions for the existence of a solution to this system in terms of ranks and Moore–Penrose inverses are provided. The general solution to the system is given when the solvability conditions are satisfied. Applications that are discussed include the solvability conditions and general solutions to some quaternion matrix equations involving \(\phi \)-Hermicity. Some examples are given to illustrate the main results.

Keywords

Matrix equations Quaternion \(\phi \)-Hermitian matrix General solution Solvability 

Mathematics Subject Classification

15A09 15A24 15B33 15B57 

Notes

Acknowledgements

The author would like to thank the anonymous referees for their valuable suggestions.

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsShanghai UniversityShanghaiPeople’s Republic of China

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