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

• Zhuo-Heng He
Article

## Abstract

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.

## Keywords

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

## Mathematics Subject Classification

15A23 15A24 15B33 15B48 15B57

## Notes

### Acknowledgements

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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