<p>In this paper, we investigate the solutions to the dual quaternion matrix equation <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$AXB = C$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mi>X</mi> <mi>B</mi> <mo>=</mo> <mi>C</mi> </mrow> </math></EquationSource> </InlineEquation>, including the general solution, the solution containing only the standard part, and the solution containing only the dual part. First, we present the real representation matrix of the dual quaternion matrix and analyze the properties of its vector operator. Then we transform the dual quaternion matrix equation into a real linear system and give the equivalent condition for the existence of the solutions to the dual quaternion matrix equation <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$AXB=C$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mi>X</mi> <mi>B</mi> <mo>=</mo> <mi>C</mi> </mrow> </math></EquationSource> </InlineEquation>, which includes the general solution, the solution containing only the standard part, and the solution containing only the dual part. Furthermore, we obtain the expressions of these solutions and the minimal <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\textrm{F}^R$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mtext>F</mtext> <mi>R</mi> </msup> </math></EquationSource> </InlineEquation>-norm solutions to the dual quaternion matrix equation <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$AXB=C$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mi>X</mi> <mi>B</mi> <mo>=</mo> <mi>C</mi> </mrow> </math></EquationSource> </InlineEquation>. Finally, we propose the algorithms and conduct numerical experiments to verify the validity of our method.</p>

Solutions of the Dual Quaternion Matrix Equation $AXB = C$

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发表日期：

2025年6月20日

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Yinping Li,
Ying Li,
Xiaochen Liu,
Ning Shao

摘要

In this paper, we investigate the solutions to the dual quaternion matrix equation $AXB = C$ $A X B = C$ , including the general solution, the solution containing only the standard part, and the solution containing only the dual part. First, we present the real representation matrix of the dual quaternion matrix and analyze the properties of its vector operator. Then we transform the dual quaternion matrix equation into a real linear system and give the equivalent condition for the existence of the solutions to the dual quaternion matrix equation $AXB=C$ $A X B = C$ , which includes the general solution, the solution containing only the standard part, and the solution containing only the dual part. Furthermore, we obtain the expressions of these solutions and the minimal $\textrm{F}^R$ $F^{R}$ -norm solutions to the dual quaternion matrix equation $AXB=C$ $A X B = C$ . Finally, we propose the algorithms and conduct numerical experiments to verify the validity of our method.

Solutions of the Dual Quaternion Matrix Equation \(AXB = C\)

摘要