On the split quaternion matrix equation \(AX=B\)

  • Xin Liu
  • Zhuo-Heng HeEmail author
Original Paper


In the paper, we consider the split quaternion matrix equation \(AX=B\). We design several real representations of split quaternion matrix to transform the above split quaternion matrix equation into some real matrix equations. By using this method, we give some necessary and sufficient conditions for \(AX=B\) to have a X or \(X=\pm X^{\star }\) solution and derive the expressions of solutions when equation is solvable, where \(X^{\star } \in \{X^*, X^\eta , X^{\eta *}\}\), \(X^*\) is the conjugate transpose of X, for \(\eta \in \{i, j, k\}\), \( X^{\eta }, X^{\eta *}\) are \(\eta \)-conjugate, \(\eta \)-Hermitian of X, respectively. We also present the solvability conditions and expression of the unique solution X or \(X=\pm X^{\star }\).


Split quaternion matrix equation Real representation Hermitian \(\eta \)-(anti-)Hermitian and \(\eta \)-(anti)-conjugate 

Mathematics Subject Classification

15A33 15A24 15B57 



This research was supported by the National Natural Science Foundation of China (Grant no. 11801354), National Natural Science Foundation for the Youth of China (Grant no. 11701598) and The Science and Technology Development Fund, Macau SAR (File no. 185/2017/A3).


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Copyright information

© Tusi Mathematical Research Group (TMRG) 2019

Authors and Affiliations

  1. 1.Faculty of Information TechnologyMacau University of Science and TechnologyTaiPaPeople’s Republic of China
  2. 2.Department of MathematicsShanghai UniversityShanghaiPeople’s Republic of China

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