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Dual quaternions have wide applications in automatic differentiation, computer graphics, mechanics, and others. Due to its application in control theory, matrix equation A X B = C has been extensively studied. However, there is currently limited information on matrix equation A X B = C regarding the dual quaternion algebra. In this paper, we provide the necessary and sufficient conditions for the solvability of dual quaternion matrix equation A X B = C , and present the expression for the general solution when it is solvable. As an application, we derive the ϕ -Hermitian solutions for dual quaternion matrix equation A X A ϕ = C , where the ϕ -Hermitian extends the concepts of Hermiticity and η -Hermiticity. Lastly, we present a numerical example to verify the main research results of this paper. [ABSTRACT FROM AUTHOR]

We propose a practical method for computing the singular value decomposition of dual quaternion matrices. The dual quaternion Householder matrix is first proposed, and by combining the properties of dual quaternions, we can implement the transformation of a dual quaternion matrix to a bidiagonalized dual number matrix. We have proven that the singular values of a dual quaternion matrix are same to the singular values of its bidiagonalized dual number matrix. Numerical experiment demonstrates the effectiveness of our proposed method for computing the singular value decomposition of dual quaternion matrices. [ABSTRACT FROM AUTHOR]