In this paper, we first study the projections onto the set of unit dual quaternions, and the set of dual quaternion vectors with unit norms. Then we propose a power method for computing the dominant eigenvalue of a dual quaternion Hermitian matrix. For a strict dominant eigenvalue, we show the sequence generated by the power method converges to the dominant eigenvalue and its corresponding eigenvector linearly. For a general dominant eigenvalue, we show the standard part of the sequence generated by the power method converges to the standard part of the dominant eigenvalue and its corresponding eigenvector linearly. Based upon these, we reformulate the simultaneous localization and mapping (SLAM) problem as a rank-one dual quaternion completion problem. A two-block coordinate descent method is proposed to solve this problem. One block has a closed-form solution and the other block is the best rank-one approximation problem of a dual quaternion Hermitian matrix, which can be computed by the power method. Numerical experiments are presented to show the efficiency of our proposed power method.

翻译：本文首先研究单位对偶四元数集合以及具有单位范数的对偶四元数向量集合上的投影问题。随后，我们提出一种用于计算对偶四元数Hermitian矩阵主特征值的幂法。对于严格主特征值，我们证明幂法生成的序列线性收敛至该主特征值及其对应特征向量。对于一般主特征值，我们证明幂法生成序列的标准部分线性收敛至主特征值的标准部分及其对应特征向量。基于此，我们将同时定位与地图构建（SLAM）问题重新表述为秩一对偶四元数补全问题，并提出一种双块坐标下降法求解该问题。其中一个块具有闭式解，另一个块则是通过对偶四元数Hermitian矩阵的最佳秩一逼近问题，可通过幂法计算。数值实验表明我们提出的幂法具有高效性。