Four symmetric systems of the matrix equations with an application over the Hamilton quaternions

Long-Sheng Liu; Shuo Zhang; Hai-Xia Chang; Long-Sheng Liu; Shuo Zhang; Hai-Xia Chang

doi:10.3934/math.20241607

AIMS Mathematics

2024, Volume 9, Issue 12: 33662-33691. doi: 10.3934/math.20241607

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Four symmetric systems of the matrix equations with an application over the Hamilton quaternions

1.
School of Mathematics and Physics, Anqing Normal University, Anqing 246011, China
2.
School of Statistics and Mathematics, Shanghai Lixin University of Accounting and Finance, Shanghai 201209, China

Received: 04 August 2024 Revised: 30 October 2024 Accepted: 12 November 2024 Published: 28 November 2024
MSC : 15A03, 15A09, 15A24

In this paper, we established some necessary and sufficient conditions for the four symmetric systems to be consistent. Moreover, we derived the expressions of their general solutions when they were solvable. As an application, we investigated the solvability conditions of matrix equations involving $ \eta $-Hermicity matrices. Finally, we presented an example to illustrate the main results of this paper.
- matrix equation,
- quaternion,
- Moore-Penrose,
- rank,
- $ \eta $-Hermitian matrix
Citation: Long-Sheng Liu, Shuo Zhang, Hai-Xia Chang. Four symmetric systems of the matrix equations with an application over the Hamilton quaternions[J]. AIMS Mathematics, 2024, 9(12): 33662-33691. doi: 10.3934/math.20241607

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Abstract

In this paper, we established some necessary and sufficient conditions for the four symmetric systems to be consistent. Moreover, we derived the expressions of their general solutions when they were solvable. As an application, we investigated the solvability conditions of matrix equations involving $ \eta $-Hermicity matrices. Finally, we presented an example to illustrate the main results of this paper.

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