Dual quaternion QR decompositon and its corresponding complex structure-preserving algorithms

The dual quaternion matrix has important application value in brain science and multi-agent formation control. In this paper, a practical method for realizing dual quaternion QR decomposition (DQQRD) is proposed by using a dual quaternion Householder transformation. Since the product of dual quaternions depends on the product law of quaternions, it will face complex computational problems. If DQQRD is directly performed, it will be inefficient. Therefore, in this paper, the complex representation of a dual quaternion matrix is established by using the semi-tensor product (STP) of matrices, and the complex structure-preserving algorithm of the DQQRD is proposed. In order to improve the accuracy of the decomposition, a method of column pivoting is given. Numerical experiments show that the method is effective. Finally, the DQQRD is applied to solve the dual quaternion linear equation \(Ax=b\) $Ax=b$ .