Solving a constrained Sylvester-type system on the commutative quaternion ring

Xiao-Xiao Ma; Long-Sheng Liu; Xiao-Quan Chen; Xiao-Xiao Ma; Long-Sheng Liu; Xiao-Quan Chen

doi:10.3934/math.20251270

AIMS Mathematics

2025, Volume 10, Issue 12: 28861-28877. doi: 10.3934/math.20251270

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Solving a constrained Sylvester-type system on the commutative quaternion ring

School of Mathematics and Physics, Anqing Normal University, Anqing 246011, China

Received: 17 October 2025 Revised: 21 November 2025 Accepted: 26 November 2025 Published: 10 December 2025
MSC : 15A09, 15A03, 15A24, 15B57

This paper establishes solvability conditions and general solution sets for a constrained Sylvester-type system over the commutative quaternion ring by proposing two distinct methods. As a key application, we also analyze the minimum solution of a related optimization problem when this system is solvable. Additionally, the solvability condition and the general form of the Hermitian solutions for this system over the commutative quaternion ring are established in this paper. The main results are validated through an algorithm and a numerical example.
- Sylvester-type system,
- commutative quaternion ring,
- Hermitian solution,
- minimum solution
Citation: Xiao-Xiao Ma, Long-Sheng Liu, Xiao-Quan Chen. Solving a constrained Sylvester-type system on the commutative quaternion ring[J]. AIMS Mathematics, 2025, 10(12): 28861-28877. doi: 10.3934/math.20251270

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Abstract

This paper establishes solvability conditions and general solution sets for a constrained Sylvester-type system over the commutative quaternion ring by proposing two distinct methods. As a key application, we also analyze the minimum solution of a related optimization problem when this system is solvable. Additionally, the solvability condition and the general form of the Hermitian solutions for this system over the commutative quaternion ring are established in this paper. The main results are validated through an algorithm and a numerical example.

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