Structure, Properties and Applications of Some Simultaneous Decompositions for Quaternion Matrices Involving \(\phi \)-Skew-Hermicity

  • Zhuo-Heng HeEmail author


Let \({\mathbb {H}}\) be the real quaternion algebra and \({\mathbb {H}}^{m\times n}\) denote the set of all \(m\times n\) matrices over \({\mathbb {H}}\). For \(A\in {\mathbb {H}}^{m\times n},\) we denote by \(A_{\phi }\) the \(n\times m\) matrix obtained by applying \(\phi \) entrywise to the transposed matrix \(A^{t},\) where \(\phi \) is a nonstandard involution of \({\mathbb {H}}\). \(A\in {\mathbb {H}}^{n\times n}\) is said to be \(\phi \)-skew-Hermitian if \(A=-A_{\phi }\). In this paper, we investigate and analyze in detail the structure and properties of a simultaneous decomposition for six quaternion matrices involving \(\phi \)-skew-Hermicity:
where A and F are \(\phi \)-skew-Hermitian. Using this simultaneous decomposition, we give some practical necessary and sufficient conditions for the existence of a \(\phi \)-skew-Hermitian solution (XYZ) to the system of quaternion matrix equations
Apart from proving an expression for the general \(\phi \)-skew-Hermitian solution to this system, we derive the \(\beta (\phi )\)-signature bounds of the \(\phi \)-skew-Hermitian solution in terms of the coefficient matrices. Moreover, we obtain necessary and sufficient conditions for the system to have \(\beta (\phi )\)-positive definite, \(\beta (\phi )\)-positive semidefinite, \(\beta (\phi )\)-negative definite and \(\beta (\phi )\)-negative semidefinite solutions. We also give some numerical examples to illustrate our results.


Quaternion Matrix decomposition Matrix equation \(\beta (\phi )\)-signature \(\phi \)-Skew-Hermitian solution 

Mathematics Subject Classification

15A23 15A24 15B33 15B48 15B57 



The authors thank Professor Rafal Ablamowicz and the anonymous referee for their valuable suggestions that led to the improvement of the presentation in this paper.


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Authors and Affiliations

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

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