<p>Dual quaternion has a wide range of applications in various fields, and the study of its matrix theory has become a hot topic in recent years. In the theoretical study of dual quaternion matrices, the <i>LU</i> decomposition plays an important role. However, due to the non-commutativity of dual quaternions, the calculation of <i>LU</i> decomposition becomes difficult. In this paper, by means of the quaternion representation of the dual quaternion matrices given by semi-tensor product of matrices and the complex representation of the quaternion matrices, we give the complex representation of the dual quaternion matrices and its properties. The complex representation we proposed greatly facilitates the simplification of computational processes. And using these properties, we propose a fast and efficient complex structure-preserving algorithm for <i>LU</i> decomposition of dual quaternion matrices. The algorithm avoids the complexity of dual quaternion operations (essentially, quaternion operations). In order to ensure the stability of the algorithm, we further give a partial pivoting dual quaternion <i>LU</i> decomposition algorithm. Based on <i>LU</i> decomposition, we also present a complex structure-preserving algorithm for <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(LDL^H\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>L</mi> <mi>D</mi> <msup> <mi>L</mi> <mi>H</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> decomposition of dual quaternion Hermitian matrices. In addition, we illustrate the effectiveness of the complex structure-preserving algorithms through numerical experiments. Finally, we give the application of dual quaternion matrix <i>LU</i> decomposition in color image authentication and kinematic linear equations.</p>