Exact and least-squares solutions of a generalized Sylvester-transpose matrix equation over generalized quaternions

Janthip Jaiprasert; Pattrawut Chansangiam; Janthip Jaiprasert; Pattrawut Chansangiam

doi:10.3934/era.2024126

Electronic Research Archive

2024, Volume 32, Issue 4: 2789-2804. doi: 10.3934/era.2024126

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Research article

Exact and least-squares solutions of a generalized Sylvester-transpose matrix equation over generalized quaternions

Janthip Jaiprasert ,
Pattrawut Chansangiam ^,

Department of Mathematics, School of Science, King Mongkut's Institute of Technology Ladkrabang, Bangkok 10520, Thailand

Received: 26 December 2023 Revised: 07 March 2024 Accepted: 29 March 2024 Published: 08 April 2024

We have considered a generalized Sylvester-transpose matrix equation $ AXB + CX^TD = E, $ where $ A, B, C, D, $ and $ E $ are given rectangular matrices over a generalized quaternion skew-field, and $ X $ is an unknown matrix. We have applied certain vectorizations and real representations to transform the matrix equation into a matrix equation over the real numbers. Thus, we have investigated a solvability condition, general exact/least-squares solutions, minimal-norm solutions, and the exact/least-squares solution closest to a given matrix. The main equation included the equation $ AXB = E $ and the Sylvester-transpose equation. Our results also covered such matrix equations over the quaternions, and quaternionic linear systems.
Citation: Janthip Jaiprasert, Pattrawut Chansangiam. Exact and least-squares solutions of a generalized Sylvester-transpose matrix equation over generalized quaternions[J]. Electronic Research Archive, 2024, 32(4): 2789-2804. doi: 10.3934/era.2024126

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Abstract

We have considered a generalized Sylvester-transpose matrix equation $ AXB + CX^TD = E, $ where $ A, B, C, D, $ and $ E $ are given rectangular matrices over a generalized quaternion skew-field, and $ X $ is an unknown matrix. We have applied certain vectorizations and real representations to transform the matrix equation into a matrix equation over the real numbers. Thus, we have investigated a solvability condition, general exact/least-squares solutions, minimal-norm solutions, and the exact/least-squares solution closest to a given matrix. The main equation included the equation $ AXB = E $ and the Sylvester-transpose equation. Our results also covered such matrix equations over the quaternions, and quaternionic linear systems.

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