<p>Dual quaternions provide a compact and efficient representation of rigid-body motions and have found widespread applications in robotics, computer vision, and control. However, their inherent algebraic complexity—stemming from the dual unit and the associated zero divisors—has hindered systematic studies of dual quaternion matrix equations. In this paper, we propose a novel real representation of dual quaternion matrices and establish an L-structured method for characterizing (anti-)(<InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\eta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>η</mi> </math></EquationSource> </InlineEquation>-)Hermitian and (anti-)(<InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\eta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>η</mi> </math></EquationSource> </InlineEquation>-)bi-Hermitian matrices. Within this unified framework, we derive necessary and sufficient conditions for the existence of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\eta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>η</mi> </math></EquationSource> </InlineEquation>-Hermitian and anti-<InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\eta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>η</mi> </math></EquationSource> </InlineEquation>-bi-Hermitian solutions to the systems <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$AXB + CXD = E$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mi>X</mi> <mi>B</mi> <mo>+</mo> <mi>C</mi> <mi>X</mi> <mi>D</mi> <mo>=</mo> <mi>E</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$(AXB, CXD) = (E, F)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mi>X</mi> <mi>B</mi> <mo>,</mo> <mi>C</mi> <mi>X</mi> <mi>D</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>E</mi> <mo>,</mo> <mi>F</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Explicit solution formulas are presented for the consistent cases, together with least-squares solutions for inconsistent systems. Finally, representative numerical examples and an application in kinematics are provided to demonstrate the effectiveness of the proposed theoretical results.</p>

L-structure method for solving special classes of solutions to dual quaternion linear matrix systems

期刊论文
发表日期：

2026年3月11日

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Zi-Han Gao,
Qing-Wen Wang

摘要

Dual quaternions provide a compact and efficient representation of rigid-body motions and have found widespread applications in robotics, computer vision, and control. However, their inherent algebraic complexity—stemming from the dual unit and the associated zero divisors—has hindered systematic studies of dual quaternion matrix equations. In this paper, we propose a novel real representation of dual quaternion matrices and establish an L-structured method for characterizing (anti-)( $\eta $ $η$ -)Hermitian and (anti-)( $\eta $ $η$ -)bi-Hermitian matrices. Within this unified framework, we derive necessary and sufficient conditions for the existence of $\eta $ $η$ -Hermitian and anti- $\eta $ $η$ -bi-Hermitian solutions to the systems $AXB + CXD = E$ $A X B + C X D = E$ and $(AXB, CXD) = (E, F)$ $(A X B, C X D) = (E, F)$ . Explicit solution formulas are presented for the consistent cases, together with least-squares solutions for inconsistent systems. Finally, representative numerical examples and an application in kinematics are provided to demonstrate the effectiveness of the proposed theoretical results.