An $ \eta $-Hermitian solution to a two-sided matrix equation and a system of matrix equations over the skew-field of quaternions

Mahmoud S. Mehany; Faizah D. Alanazi; Mahmoud S. Mehany; Faizah D. Alanazi

doi:10.3934/math.2025352

AIMS Mathematics

2025, Volume 10, Issue 4: 7684-7705. doi: 10.3934/math.2025352

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An $ \eta $-Hermitian solution to a two-sided matrix equation and a system of matrix equations over the skew-field of quaternions

Mahmoud S. Mehany ^{1,2
,
,},
Faizah D. Alanazi ³

1.
Department of Mathematics, Faculty of Education, Ain Shams University, Cairo, 11566, Egypt
2.
School of Science and Technology, NOVA University, Lisbon, Cairo Campus, New Cairo, 2024, Egypt
3.
Department of Mathematics, College of Science, Northern Border University, Arar, Saudi Arabia

Received: 15 January 2025 Revised: 11 March 2025 Accepted: 17 March 2025 Published: 01 April 2025
MSC : 15A05, 15A09, 15A36, 15A63, 68W30

This study investigated $ {\eta} $-Hermitian solutions to a two-sided matrix equation and a system of matrix equations over the skew-field of quaternions. We employed solvability conditions to identify the general solution for this system. This method deduces the Moore-Penrose inverse, and the rank equalities for the system's coefficients. We utilized these strategies to develop an algorithm that can compute general solutions. We also used these algorithms in numerical instances to verify the theoretical results.
- linear matrix equations,
- MATLAB software,
- nonstandard involution,
- generalized inverses,
- matrix rank
Citation: Mahmoud S. Mehany, Faizah D. Alanazi. An $ \eta $-Hermitian solution to a two-sided matrix equation and a system of matrix equations over the skew-field of quaternions[J]. AIMS Mathematics, 2025, 10(4): 7684-7705. doi: 10.3934/math.2025352

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Abstract

This study investigated $ {\eta} $-Hermitian solutions to a two-sided matrix equation and a system of matrix equations over the skew-field of quaternions. We employed solvability conditions to identify the general solution for this system. This method deduces the Moore-Penrose inverse, and the rank equalities for the system's coefficients. We utilized these strategies to develop an algorithm that can compute general solutions. We also used these algorithms in numerical instances to verify the theoretical results.

References

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